A vector $\vec{a}=\alpha \hat{i}+2 \hat{j}+\beta \hat{k}(\alpha, \beta \in R)$ lies in the plane of the vectors, $\vec{b}=\hat{i}+\hat{j}$ and $\vec{c}=\hat{i}-\hat{j}+4 \hat{k}$. If $\vec{a}$ bisects the angle between $\vec{b}$ and $\vec{c}$, then:

(B) $\overrightarrow{\mathrm{a}} \cdot \hat{\mathrm{k}}+2=0$

(A) $\vec{a} \cdot \hat{k}+4=0$

(C) $\overrightarrow{\mathrm{a}} \cdot \hat{\mathrm{i}}+3=0$

(D) $\vec{a} \cdot \hat{i}+1=0$

JEE Main · 2020 · 07 Jan 2020 · Shift 1 · Vector Algebra

A vector

a = α \hat{i} + 2 \hat{j} + β \hat{k} (α, β \in R)

lies in the plane of the vectors,

b = \hat{i} + \hat{j}

and

c = \hat{i} - \hat{j} + 4 \hat{k}

. If

a

bisects the angle between

b

and

c

, then:

Answer: (B)

Write

\hat{b} = \frac{i ^ + j ^}{2}

and

\overset{c}{^} = \frac{i ^ - j ^ + 4 k ^}{3 2}

, since

∣ b ∣ = 2

and

∣ c ∣ = 32

. A bisector of the angle between them is parallel to

\hat{b} \pm \overset{c}{^}

; the combination

\hat{b} - \overset{c}{^} \propto \hat{i} + 2 \hat{j} - 2 \hat{k}

already carries its components in the ratio required by

a = α \hat{i} + 2 \hat{j} + β \hat{k}

. Scaling it so the

\hat{j}

component is

2

gives

a = \hat{i} + 2 \hat{j} - 2 \hat{k}

, whose

\hat{k}

component is

- 2

, so

a \cdot \hat{k} + 2 = 0

JEE Main · 2020 · 07 Jan 2020 · Shift 1 · Single correct option