The vector $\vec{a}$ and $\vec{b}$ are not perpendicular and $\vec{c}$ and $\vec{d}$ are two vectors satisfying: $\vec{b} \times \vec{c}=\vec{b} \times \vec{d}$ and $\vec{a} \cdot \vec{d}=0$. Then the vector $\vec{d}$ is equal to

(C) $\vec{c}-\left(\frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{b}$

(A) $\vec{c}+\left(\frac{\vec{a} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{b}$

(B) $\vec{b}+\left(\frac{\vec{b} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{c}$

(D) $\vec{b}-\left(\frac{\vec{b} \cdot \vec{c}}{\vec{a} \cdot \vec{b}}\right) \vec{c}$

JEE Main · 2011 · 10 Apr 2011 · Offline · Vector Algebra

The vector

a

and

b

are not perpendicular and

c

and

d

are two vectors satisfying:

b \times c = b \times d

and

a \cdot d = 0

. Then the vector

d

is equal to

Answer: (C)

From

b \times c = b \times d

, cross both sides with

a

a \times (b \times c) = a \times (b \times d)

. The expansion

a \times (b \times c) = (a \cdot c) b - (a \cdot b) c

turns this into

(a \cdot c) b - (a \cdot b) c = (a \cdot d) b - (a \cdot b) d

. Since

a \cdot d = 0

, the first term on the right drops out, leaving

(a \cdot b) d = (a \cdot b) c - (a \cdot c) b

, so

d = c - \frac{a \cdot c}{a \cdot b} b

JEE Main · 2011 · 10 Apr 2011 · Offline · Single correct option