Let \overset{\rightarrow}{a} \textrm{ } and \textrm{ } \overset{\rightarrow}{b} be two unit vectors such that \overset{\rightarrow}{a} \textrm{ } \cdot \textrm{ } \overset{\rightarrow}{b} \textrm{ } = \textrm{ } 0 . For some x, y \in R, let \overset{\rightarrow}{a} \textrm{ } \cdot \textrm{ } \overset{\rightarrow}{b} \textrm{ } = \textrm{ } 0 . If \left|\right. \overset{\rightarrow}{c} \left|\right. = 2 and the vector \overset{\rightarrow}{c} is inclined at the same angle a to both \overset{\rightarrow}{a} \textrm{ } and \textrm{ } \overset{\rightarrow}{b}, then the value of 8cos2α is ______

Engineering

Mathematics

Vector Product of Two Vectors

Question

Let $\vec{a} and \vec{b}$ be two unit vectors such that $\vec{a} \cdot \vec{b} = 0 .$ For some x, y $\in$ R, let $\vec{a} \cdot \vec{b} = 0 .$ If $| \vec{c} | = 2$ and the vector $\vec{c}$ is inclined at the same angle a to both $\vec{a} and \vec{b}$ , then the value of 8cos²α is _______.

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$\bar{a} \cdot \bar{c} = x = 2 cos α$

$\bar{b} \cdot \bar{c} = y = 2 cos α$

$\bar{c} = (2 cosα) \bar{a} + (2 cosα) \bar{b} + (\bar{a} \times \bar{b})$

$| \bar{c} |$ = 4 cos² α + 4 cos²α + 1 = 4 ⇒ 8 cos²α = 3