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

Mathematics · Vector Algebra

JEE Advanced 2018 — Paper 1 — Question 38

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 $\vec{c}=x \vec{a}+y \vec{b}+(\vec{a} \times \vec{b})$ . If $|\vec{c}|=2$ and the vector $\vec{c}$ is inclined at the same angle $\alpha$ to both $\vec{a}$ and $\vec{b}$ , then the value of $8 \cos ^{2} \alpha$ is $\_\_\_\_$ .

Answer: 3

Numerical answer — enter this value.

Step-by-step solution

$\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{a}}=\mathrm{x} \Rightarrow 2 \cos \alpha=\mathrm{x}$

$\overrightarrow{\mathrm{c}} \cdot \overrightarrow{\mathrm{b}}=\mathrm{y} \Rightarrow 2 \cos \alpha=\mathrm{y}$

$\because|\overrightarrow{\mathrm{c}}|=2$

$\therefore x^{2}+y^{2}+1=4 \Rightarrow 8 \cos ^{2} \alpha=3$

( $\because \overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}$ all are $\perp$ to each other $\&$ are unit vectors)

Answer key and solution verified before publishing.

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Exam: JEE Advanced 2018
Paper: Paper 1
Subject: Mathematics
Chapter: Vector Algebra
Topic: Scalar or Dot Product of Two Vectors