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4.6 Establish the following vector inequalities geometrically or otherwise:
(a) |a+b| < |a| + |b|
(b) |a+b| > ||a| −|b||
(c) |a−−b| < |a| + |b|
(d) |a−−b| > ||a| − |b||
When does the equality sign above apply?
4.6 Establish the following vector inequalities geometrically or otherwise:
(a) |a+b| < |a| + |b|
(b) |a+b| > ||a| −|b||
(c) |a−−b| < |a| + |b|
(d) |a−−b| > ||a| − |b||
When does the equality sign above apply?
-
1 Answer
-
(a)
We can write
=
= = ………(ii)
=
In a triangle, each side is smaller than the sum of the other two sides. Therefore in ΔOMN, we have ON < (OM + MN)
< + ….(iv)
If the two vectors and act along a straight line in the same direction, then we can write = +
Combining equations (iv) and (v), we get
+
(b)
Let two vectors and be represented by the adjacent sides of a parallelogram OMNP.
We can write
=
= =
...more(a)
We can write
=
= = ………(ii)
=
In a triangle, each side is smaller than the sum of the other two sides. Therefore in ΔOMN, we have ON < (OM + MN)
< + ….(iv)
If the two vectors and act along a straight line in the same direction, then we can write = +
Combining equations (iv) and (v), we get
+
(b)
Let two vectors and be represented by the adjacent sides of a parallelogram OMNP.
We can write
=
= = ………(ii)
=
In a triangle, each side is smaller than the sum of the other two sides. Therefore in ΔOMN, we have ON + MN > OM and ON + OM > MN
- ……(iv)
If the two vectors and act along a straight line in the same direction, then we can write = -
Combining (iv) and (v), we get:
-
(c)
Let two vectors and be represented by the adjacent sides of a parallelogram PORS.
We can write
= = ………(i)
=
In a triangle, each side is smaller than the sum of the other two sides. Therefore in ΔOPS, we have OS < OP + PS
+
+ …….(iii)
If the two vectors and act along a straight line but in the opposite direction, then we can write = +
Combining (iii) and (iv), we get
+
(d)
We can write
OS + PS > OP …….(i)
OS > PS – OP …….(ii)
+ ….(iii)
The quantity on the LHS is always positive and that on the RHS can be positive or negative. To make both quantities positive, we take modulus of both sides as:
> ….(iv)
If the two vectors and act along a straight line but in the opposite direction, then we can write = -
Combining (iv) and (v), we get
-
less<p><strong>(a)</strong></p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1728970696php8INBOC_480x360.jpeg" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1728970696php8INBOC.jpeg" alt="" width="519" height="292"></picture></div></div><p>We can write</p><p><span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>O</mi><mi>M</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced><mo>…</mo><mo>…</mo><mo>…</mo><mo>.</mo><mfenced separators="|"><mrow><mrow><mi mathvariant="normal">i</mi></mrow></mrow></mfenced></math></span></p><p><math><mi></mi></math> <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>M</mi><mi>N</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> = <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>O</mi><mi>P</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover></math></span> ………(ii)</p><p><math><mi></mi></math> <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>O</mi><mi>N</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>+</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced><mo>…</mo><mo>…</mo><mo>.</mo><mfenced separators="|"><mrow><mrow><mi>i</mi><mi>i</mi><mi>i</mi></mrow></mrow></mfenced></math></span></p><p>In a triangle, each side is smaller than the sum of the other two sides. Therefore in ΔOMN, we have ON < (OM + MN)</p><p><span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>+</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></math></span> < <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> + <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> ….(iv)</p><p>If the two vectors <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mi></mi></math></span> and <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></math></span> act along a straight line in the same direction, then we can write <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>+</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> + <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced><mo>…</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>v</mi><mo>)</mo></math></span></p><p>Combining equations (iv) and (v), we get</p><p><span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>+</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></math></span> <span title="Click to copy mathml"><math><mo>≤</mo></math></span> <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> + <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span></p><p><strong>(b)</strong></p><div><div><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1728971015phposuCW7_480x360.jpeg" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1728971015phposuCW7.jpeg" alt="" width="253" height="160"></picture></div></div><p> </p></div></div><p>Let two vectors <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></math></span> and <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></math></span> be represented by the adjacent sides of a parallelogram OMNP.</p><p>We can write</p><p><span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>O</mi><mi>M</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced><mo>…</mo><mo>…</mo><mo>…</mo><mo>.</mo><mfenced separators="|"><mrow><mrow><mi mathvariant="normal">i</mi></mrow></mrow></mfenced></math></span></p><p><math><mi></mi></math> <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>M</mi><mi>N</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> = <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>O</mi><mi>P</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover></math></span> ………(ii)</p><p><math><mi></mi></math> <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>O</mi><mi>N</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>+</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced><mo>…</mo><mo>…</mo><mo>.</mo><mfenced separators="|"><mrow><mrow><mi>i</mi><mi>i</mi><mi>i</mi></mrow></mrow></mfenced></math></span></p><p>In a triangle, each side is smaller than the sum of the other two sides. Therefore in ΔOMN, we have ON + MN > OM and ON + OM > MN</p><p><math><mi></mi></math> <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>O</mi><mi>N</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover><mo>></mo><mi></mi><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>O</mi><mi>M</mi><mo>-</mo><mi>O</mi><mi>P</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span></p><p><span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>+</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced><mo>></mo></math></span> <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> - <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced><mi></mi></math></span> ……(iv)</p><p>If the two vectors <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mi></mi></math></span> and <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></math></span> act along a straight line in the same direction, then we can write <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>+</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> - <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced><mo>…</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>v</mi><mo>)</mo></math></span></p><p>Combining (iv) and (v), we get:</p><p><span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>+</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced><mo>≥</mo></math></span> <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> - <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced><mi></mi></math></span></p><p><strong>(c)</strong></p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1728970992phpfd6Ygl_480x360.jpeg" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1728970992phpfd6Ygl.jpeg" alt="" width="207" height="154"></picture></div><div><p>Let two vectors <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></math></span> and <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></math></span> be represented by the adjacent sides of a parallelogram PORS.</p><p>We can write</p><p><math><mi></mi></math> <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>O</mi><mi>R</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> = <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>P</mi><mi>S</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover></math></span> ………(i)</p><p><span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mi>O</mi><mi>P</mi></mrow></mrow></mfenced></mrow></mrow><mo>?</mo></mover></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced><mo>…</mo><mo>…</mo><mo>…</mo><mo>.</mo><mfenced separators="|"><mrow><mrow><mi mathvariant="normal">i</mi><mi mathvariant="normal">i</mi></mrow></mrow></mfenced></math></span></p><p>In a triangle, each side is smaller than the sum of the other two sides. Therefore in ΔOPS, we have OS < OP + PS</p><p><span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>-</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></math></span> <span title="Click to copy mathml"><math><mo><</mo></math></span> <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> + <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mo>-</mo><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span></p><p><span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>-</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></math></span> <span title="Click to copy mathml"><math><mo><</mo></math></span> <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> + <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> …….(iii)</p><p>If the two vectors <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mi></mi></math></span> and <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></math></span> act along a straight line but in the opposite direction, then we can write <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>-</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> + <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced><mo>…</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>i</mi><mi>v</mi><mo>)</mo></math></span></p><p>Combining (iii) and (iv), we get</p><p><span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>-</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></math></span> <span title="Click to copy mathml"><math><mo>≤</mo></math></span> <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> + <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span></p><p><strong>(d)</strong></p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1728971091php85fXyf_480x360.jpeg" media="(max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1728971091php85fXyf.jpeg" alt="" width="207" height="154"></picture></div></div><p>We can write</p><p>OS + PS > OP …….(i)</p><p>OS > PS – OP …….(ii)</p><p><span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>-</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></math></span> <span title="Click to copy mathml"><math><mo>></mo></math></span> <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> + <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> ….(iii)</p><p>The quantity on the LHS is always positive and that on the RHS can be positive or negative. To make both quantities positive, we take modulus of both sides as:</p><p><span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>-</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></mrow></mrow></mfenced></math></span> > <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced><mo>-</mo><mi mathvariant="normal"></mi><mi mathvariant="normal"></mi><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced><mi mathvariant="normal"></mi></mrow></mrow></mfenced><mi></mi></math></span> ….(iv)</p><p>If the two vectors <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mi></mi></math></span> and <span title="Click to copy mathml"><math><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></math></span> act along a straight line but in the opposite direction, then we can write <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>-</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></math></span> = <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> - <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced><mo>…</mo><mo>.</mo><mo>.</mo><mo>(</mo><mi>v</mi><mo>)</mo></math></span></p><p>Combining (iv) and (v), we get</p><p><span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover><mo>-</mo><mi></mi><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover><mi></mi></mrow></mrow></mfenced></math></span> <span title="Click to copy mathml"><math><mo>≥</mo></math></span> <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>a</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span> - <span title="Click to copy mathml"><math><mfenced open="|" close="|" separators="|"><mrow><mrow><mover accent="true"><mrow><mrow><mi>b</mi></mrow></mrow><mo>?</mo></mover></mrow></mrow></mfenced></math></span></p></div></div>
Similar Questions for you
Please find the solution below:
after 10 kicks,
v? = 3tî v? = 24cos 60°î + 24sin 60°? = 12î + 12√3?
v? = v? – v? = (12 – 3t)î + 12√3?
It is minimum when 12 - 3t = 0 ⇒ t = 4sec
ω = θ² + 2θ
α = (ωdω)/dθ = (θ² + 2θ) (2θ + 2)
At θ = 1rad.
ω = 3rad/s and α = 12rad/s²
a? = αR = 12 m/s² a? = ω²R = 9 m/s² A? = √ (a? ² + a? ²) = 15 m/s²
a? = v? ²/4r
a_A? = (v? ²/r²) × r = v? ²/r
a_A = 3v? ²/4r
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