SATHEE: Vectors 4 Question 2

Vectors 4 Question 2

2. Let $a, b$ and $c$ be three unit vectors, out of which vectors $b$ and $c$ are non-parallel. If $α$ and $β$ are the angles which vector $a$ makes with vectors $b$ and $c$ respectively and $a \times (b \times c) = \frac{1}{2} b$ , then $| α - β |$ is equal to

(2019 Main, 12 Jan II)

(a) $30^{\circ}$

(b) $45^{\circ}$

(d) $60^{\circ}$

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Answer:

Correct Answer: 2. (a)

Solution:

Given, $a \times (b \times c) = \frac{1}{2} b \Rightarrow (a \cdot c) b - (a \cdot b) c = \frac{1}{2} b$

$[∵ a \times (b \times c) = (a \cdot c) b - (a \cdot b) c]$

On comparing both sides, we get

$a \cdot c = \frac{1}{2}$

and $a \cdot b = 0$

$∵ a, b$ and $c$ are unit vectors, and angle between $a$ and $b$ is $α$ and angle between $a$ and $c$ is $β$ , so

$\begin{aligned} | a | | c | \cos β = \frac{1}{2} \\ [from Eq. (i)] \\ \Rightarrow \cos β = \frac{1}{2} [∵ | a | = 1 = | c |] \\ \Rightarrow β = \frac{π}{3} …(iii) [∵ \cos \frac{π}{3} = \frac{1}{2}] \end{aligned}$

and $| a | | b | \cos α = 0$ [from Eq. (ii)]

$\Rightarrow α = \frac{π}{2}$

From Eqs. (iii) and (iv), we get

$| α - β | = \frac{π}{2} - \frac{π}{3} = \frac{π}{6} = 30^{\circ}$

Vectors 4 Question 2

2. Let $a, b$ and $c$ be three unit vectors, out of which vectors $b$ and $c$ are non-parallel. If $α$ and $β$ are the angles which vector $a$ makes with vectors $b$ and $c$ respectively and $a \times (b \times c) = \frac{1}{2} b$ , then $| α - β |$ is equal to

Answer:

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Vectors 4 Question 2

2. Let a,b and c be three unit vectors, out of which vectors b and c are non-parallel. If α and β are the angles which vector a makes with vectors b and c respectively and a×(b×c)=12b, then |α−β| is equal to

Answer:

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2. Let $a, b$ and $c$ be three unit vectors, out of which vectors $b$ and $c$ are non-parallel. If $α$ and $β$ are the angles which vector $a$ makes with vectors $b$ and $c$ respectively and $a \times (b \times c) = \frac{1}{2} b$ , then $| α - β |$ is equal to