If $\vec{a}, \vec{b}, \vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a}=\alpha \vec{b}-\hat{n}$, $(\alpha \neq 0)$ and $\vec{b} \cdot \vec{c}=12$, then $|\vec{c} \times(\vec{a} \times \vec{b})|$ is equal to:

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Vector Algebra

a, b, c

are three non-zero vectors and

\overset{n}{^}

is a unit vector perpendicular to

c

such that

a = α b - \overset{n}{^}

(α \neq = 0)

and

b \cdot c = 12

, then

∣ c \times (a \times b) ∣

is equal to:

Answer: (C)

Since

\overset{n}{^} ⊥ c

, we have

\overset{n}{^} \cdot c = 0

, so from

a = α b - \overset{n}{^}

a \cdot c = α (b \cdot c) - \overset{n}{^} \cdot c = 12 α

. The vector triple product gives

c \times (a \times b) = (c \cdot b) a - (c \cdot a) b = 12 a - 12 α b = 12 (a - α b) = - 12 \overset{n}{^} .

\overset{n}{^}

is a unit vector, the magnitude is

∣ c \times (a \times b) ∣ = 12.

JEE Main · 2023 · 30 Jan 2023 · Shift 1 · Single correct option