Let a- and b- be two unit vectors such that a-b-0- For some x- y - - let c-xa-yb-a-b- If -c-2 and the vector c- is inclined at the same angle - to both a- and b- then the value of 8cos2- is -

c⃗=xa⃗+yb⃗+a⃗×b⃗ c⃗·a⃗=x and x=2cosα c⃗·b⃗=y and y=2cosα Also, |a⃗×b⃗|=1 ∴c⃗=(2cosα )(a⃗+b⃗)+a⃗×b⃗ ∴ |c⃗|^2=4cos^2α(a⃗ ...

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Byju's Answer

Standard XII

Mathematics

Dot Product of Two Vectors

Let a⃗ and ...

Question

Let $\to a$ and $\to b$ be two unit vectors such that $\to a \cdot \to b = 0$ . For some x, y $ϵ R$ , let $\to c = x \to a + y \to b + (\to a \times \to b)$ . If $| \to c | = 2$ and the vector $\to c$ is inclined at the same angle $α$ to both $\to a$ and $\to b$ , then the value of $8 {cos}^{2} α$ is __________.

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Solution

$\to c = x \to a + y \to b + \to a \times \to b$
$\to c \cdot \to a = x$ and $x = 2 cos α$
$\to c \cdot \to b = y$ and $y = 2 cos α$
Also, $| \to a \times \to b | = 1$

$∴ \to c = (2 cos α) (\to a + \to b) + \to a \times \to b$

$∴ | \to c |^{2} = 4 {cos}^{2} α (\to a + \to b)^{2} + (\to a \times \to b)^{2} + 2 cos α (\to a + \to b) \cdot (\to a \times \to b)$

$∴ 4 = 8 {cos}^{2} α + 1$

$∴ 8 {cos}^{2} α = 3$

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Let →a and →b be two unit vectors such that →a⋅→b=0. For some x, y ϵ R, let →c=x→a+y→b+(→a×→b). If |→c|=2 and the vector →c is inclined at the same angle α to both →a and →b, then the value of 8cos2α is __________.

→c=x→a+y→b+→a×→b→c⋅→a=x and x=2cosα→c⋅→b=y and y=2cosαAlso, |→a×→b|=1∴→c=(2cosα)(→a+→b)+→a×→b∴|→c|2=4cos2α(→a+→b)2+(→a×→b)2+2cosα(→a+→b)⋅(→a×→b)∴4=8cos2α+1∴8cos2α=3