4.7 Given a + b + c + d = 0, which of the following statements are correct:

(a) a, b, c, and d must each be a null vector

(b) The magnitude of (a + c) equals the magnitude of (b + d)

(c) The magnitude of a can never be greater than the sum of the magnitudes of b, c, and d

(d) b + c must lie in the plane of a and d if a and d are not collinear, and in the line of a and d, if they are collinear?

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Vishal Baghel | Contributor-Level 10

5 months ago

4.7

(a) This statement is In order to make a + b+ c + d = 0, it is not necessary to have all four given vectors to be null vectors.

(b) a + b + c +d = 0, ( a + c ) = - (b + d)

Taking the magnitude of both LHS and RHS, | a + c| = | -(b + d)| = | b + d|

So the statement is True.

(c) Given a + b + c + d = 0, a = - (b + c + d)

Taking magnitude of both LHS and RHS, |a| = | -( b + c+ d)| = |b + c +d|

|a| $\leq |b| + |c| + | d |$ ……….(1)

Equation (1) shows that the magnitude of a can never be greater than the sum of magnitudes of b,c and d. So the statement is True.

(d) In the given expression a + b + c + d = 0, the sum of 3 vectors

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4.7

(a) This statement is In order to make a + b+ c + d = 0, it is not necessary to have all four given vectors to be null vectors.

(b) a + b + c +d = 0, ( a + c ) = - (b + d)

Taking the magnitude of both LHS and RHS, | a + c| = | -(b + d)| = | b + d|

So the statement is True.

(c) Given a + b + c + d = 0, a = - (b + c + d)

Taking magnitude of both LHS and RHS, |a| = | -( b + c+ d)| = |b + c +d|

|a| $\leq |b| + |c| + | d |$ ……….(1)

Equation (1) shows that the magnitude of a can never be greater than the sum of magnitudes of b,c and d. So the statement is True.

(d) In the given expression a + b + c + d = 0, the sum of 3 vectors a, (b+c), d can be zero only if (b+c) lie in a plane containing a and d, assuming that these three vectors are represented by the three sides of a triangle. If a and d are collinear , then it implies that the vector (b+c) is in the line of a and d. This statement holds good only then the sum of all three vectors will be zero. The statement is True.

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