Let <math> <mrow> <mover accent="true"> <mrow> <mi>a</mi> </mrow> <mo stretchy="true">→</mo> </mover> <mo>=</mo> <mi>α</mi> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>−</mo> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> <mtext> </mtext> <mtext> </mtext> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mover accent="true"> <mrow> <mi>b</mi> </mrow> <mo stretchy="true">→</mo> </mover> <mo>=</mo> <mn>2</mn> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>−</mo> <mi>α</mi> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>α</mi> <mo>></mo> <mn>0</mn> <mo>.</mo> </mrow> </math> If the projection of <math> <mrow> <mover accent="true"> <mrow> <mi>a</mi> </mrow> <mo stretchy="true">→</mo> </mover> <mtext> </mtext> <mtext> </mtext> <mo>×</mo> <mtext> </mtext> <mtext> </mtext> <mover accent="true"> <mrow> <mi>b</mi> </mrow> <mo stretchy="true">→</mo> </mover> </mrow> </math> on the vector <math> <mrow> <mo>−</mo> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mn>2</mn> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>−</mo> <mn>2</mn> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> </math> is 30, then a equal to:

Given : a → = ( α , 1 , − 1 ) a n d b → = ( 2 , 1 , − α ) c → = a → × b → = | i ^ j ^ k ^ α 1 − 1 2 1 − α | = ( − α + 1 ) i ^ + ( α 2 − 2 ) j ^ + ( α − 2 ) k ^ Projection of c → on d → = − i ^ + 2 j ^ − 2 k ^ = | c → . d → | d → | | = 3 0 { G i v e n } ⇒ | α − 1 − 4 + 2 α 2 − 2 α + 4 1 + 4 + 4 | = 3 0 On solving Þ α = − 1 3 2 (Rejected as a > 0) and a = 7

Let <math> <mrow> <mover accent="true"> <mrow> <mi>a</mi> </mrow> <mo stretchy="true">→</mo> </mover> <mo>=</mo> <mi>α</mi> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>−</mo> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> <mtext> </mtext> <mtext> </mtext> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mover accent="true"> <mrow> <mi>b</mi> </mrow> <mo stretchy="true">→</mo> </mover> <mo>=</mo> <mn>2</mn> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>−</mo> <mi>α</mi> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> <mo>,</mo> <mtext> </mtext> <mtext> </mtext> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> <mo>.</mo> </mrow> </math> If the projection of <math> <mrow> <mover accent="true"> <mrow> <mi>a</mi> </mrow> <mo stretchy="true">→</mo> </mover> <mtext> </mtext> <mtext> </mtext> <mo>×</mo> <mtext> </mtext> <mtext> </mtext> <mover accent="true"> <mrow> <mi>b</mi> </mrow> <mo stretchy="true">→</mo> </mover> </mrow> </math> on the vector <math> <mrow> <mo>−</mo> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mn>2</mn> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>−</mo> <mn>2</mn> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> </math> &nbsp;is 30, then a equal to:

Ask & Answer Question

Maths Class 12th Vector Algebra

Let $\vec{a} = α \hat{i} + \hat{j} - \hat{k} a n d \vec{b} = 2 \hat{i} + \hat{j} - α \hat{k}, α > 0 .$ If the projection of $\vec{a} \times \vec{b}$ on the vector $- \hat{i} + 2 \hat{j} - 2 \hat{k}$ is 30, then a equal to:

Option 1 -

$\frac{15}{2}$

Option 2 -

Option 3 -

$\frac{13}{2}$

Option 4 -

2 Views | Posted 2 weeks ago

Asked by Shiksha User

1 Answer

Answered by

Raj Pandey | Contributor-Level 9

2 weeks ago

Correct Option - 4

Detailed Solution:

Given : $\vec{a} = (α, 1, - 1) a n d \vec{b} = (2, 1, - α) \vec{c} = \vec{a} \times \vec{b} = | \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ α & 1 & - 1 \\ 2 & 1 & - α \end{matrix} |$

$= (- α + 1) \hat{i} + (α^{2} - 2) \hat{j} + (α - 2) \hat{k}$ Projection of $\vec{c}$ on $\vec{d} = - \hat{i} + 2 \hat{j} - 2 \hat{k} = | \vec{c} . \frac{\vec{d}}{| \vec{d} |} | = 30 {G i v e n}$

$\Rightarrow | \frac{α - 1 - 4 + 2 α^{2} - 2 α + 4}{\sqrt[]{1 + 4 + 4}} | = 30$

On solving Þ $α = \frac{- 13}{2}$ (Rejected as a > 0) and a = 7

Given : <math> <mrow> <mover accent="true"> <mi>a</mi> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mtext> </mtext> <mtext> </mtext> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mover accent="true"> <mi>b</mi> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mover accent="true"> <mi>c</mi> <mo>→</mo> </mover> <mo>=</mo> <mover accent="true"> <mi>a</mi> <mo>→</mo> </mover> <mo>×</mo> <mover accent="true"> <mi>b</mi> <mo>→</mo> </mover> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mtable> <mtr columnalign="center"> <mtd columnalign="center"> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> </mtd> <mtd columnalign="center"> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> </mtd> <mtd columnalign="center"> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mi>α</mi> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr columnalign="center"> <mtd columnalign="center"> <mn>2</mn> </mtd> <mtd columnalign="center"> <mn>1</mn> </mtd> <mtd columnalign="center"> <mo>−</mo> <mi>α</mi> </mtd> </mtr> </mtable> </mrow> <mo>|</mo> </mrow> </mrow> </math>   <math> <mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>α</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mi>α</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>α</mi> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> </mrow> </math> Projection of <math> <mrow> <mover accent="true"> <mi>c</mi> <mo>→</mo> </mover> </mrow> </math> on <math> <mrow> <mover accent="true"> <mi>d</mi> <mo>→</mo> </mover> <mo>=</mo> <mo>−</mo> <mover accent="true"> <mi>i</mi> <mo>^</mo> </mover> <mo>+</mo> <mn>2</mn> <mover accent="true"> <mi>j</mi> <mo>^</mo> </mover> <mo>−</mo> <mn>2</mn> <mover accent="true"> <mi>k</mi> <mo>^</mo> </mover> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>c</mi> <mo>→</mo> </mover> <mo>.</mo> <mfrac> <mrow> <mover accent="true"> <mi>d</mi> <mo>→</mo> </mover> </mrow> <mrow> <mrow> <mo>|</mo> <mrow> <mover accent="true"> <mi>d</mi> <mo>→</mo> </mover> </mrow> <mo>|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>3</mn> <mn>0</mn> <mrow> <mo>{</mo> <mrow> <mi>G</mi> <mi>i</mi> <mi>v</mi> <mi>e</mi> <mi>n</mi> </mrow> <mo>}</mo> </mrow> </mrow> </math> <math> <mrow> <mo>⇒</mo> <mrow> <mo>|</mo> <mrow> <mfrac> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mn>4</mn> <mo>+</mo> <mn>2</mn> <msup> <mrow> <mi>α</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>α</mi> <mo>+</mo> <mn>4</mn> </mrow> <mrow> <mroot> <mrow> <mn>1</mn> <mo>+</mo> <mn>4</mn> <mo>+</mo> <mn>4</mn> </mrow> <mrow></mrow> </mroot> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mn>3</mn> <mn>0</mn> </mrow> </math>  On solving Þ <math> <mrow> <mi>α</mi> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> </mrow> </math>  (Rejected as a > 0) and a = 7

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Let $\vec{a} = α \hat{i} + \hat{j} - \hat{k} a n d \vec{b} = 2 \hat{i} + \hat{j} - α \hat{k}, α > 0 .$ If the projection of $\vec{a} \times \vec{b}$ on the vector $- \hat{i} + 2 \hat{j} - 2 \hat{k}$ is 30, then a equal to:

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Let a → = α i ^ + j ^ − k ^ a n d b → = 2 i ^ + j ^ − α k ^ , α > 0 . If the projection of a → × b → on the vector − i ^ + 2 j ^ − 2 k ^ is 30, then a equal to:

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Let $\vec{a} = α \hat{i} + \hat{j} - \hat{k} a n d \vec{b} = 2 \hat{i} + \hat{j} - α \hat{k}, α > 0 .$ If the projection of $\vec{a} \times \vec{b}$ on the vector $- \hat{i} + 2 \hat{j} - 2 \hat{k}$ is 30, then a equal to: