Question

A class XII student appearing for a competitive examination was asked to attempt the following questions.

Let  \(\vec a,\vec b\) and \(\vec c\) be three non zero vectors.

1. If  \(\vec a\) and  \(\vec b\) are such that \(|\vec a+\vec b|=|\vec a-\vec b|\) then

a. \(\vec a\perp\vec b\)

b. \(\vec a||\vec b\) 

c. \(\vec a=\vec b\)

d. None of these

2. If \(\vec a=\hat i-2\hat j,\) \(\vec b=2\hat i+\hat j+3\hat k\) then evaluate \((2\vec a+\vec b).[(2\vec a+\vec b)\times(\vec a-2\vec b)]\)

a. 0

b. 4

c. 3

d. 2

3. If  \(\vec a\) and  \(\vec b\)are unit vectors and θ be the angle between them then \(|\vec a-\vec b|\) is

a. \(sin\cfrac{\theta}2\)

b. 2 \(sin\cfrac{\theta}2\)

c. 2 \(cos\cfrac{\theta}2\)

d. \(cos\cfrac{\theta}2\)

4. Let \(\vec a,\,\vec b\) and  \(\vec c\) be unit vectors such that \(\vec a.\vec b=\vec a.\vec c=0\) and angle between \(\vec b\) and  \(\vec c\) is \(\cfrac{\pi}6\) then \(\vec a\) =

a. 2(\(\vec b\times\vec c\)). 

b. -2(\(\vec b\times\vec c\)) 

c. ±2(\(\vec b\times\vec c\)) 

d. 2(\(\vec b\pm\vec c\))

5. The area of the parallelogram formed by \(\vec a\) and \(\vec b\) as diagonals is

a. 70

b. 35

c. \(\cfrac{\sqrt{70}}2\)

d. \(\sqrt{70}\) 

Answer 1

1. (a) |\(\vec a+\vec b\) |² = |\(\vec a-\vec b\)|² ⇒ 2.\(\vec a.\vec b\)= 0, \(\vec a\perp\vec b\)

2. (a) 0

3. (b) 2 \(sin\cfrac{\theta}2\)

4. (c) ±2(\(\vec b\times\vec c\)) 

5. (c) √70/2 sq units