Let a=2 i- j+ k, b= i+2 j- k and c= i+ j-2 k be three vectors.

Question

Let $\vec a=2\hat i-\hat j+\hat k$, $\vec b=\hat i+2\hat j-\hat k$ and $\vec c=\hat i+\hat j-2\hat k$ be three vectors. A vector of the type $\vec b+\lambda\vec c$ for some scalar $\lambda$, whose projection on $\vec a$ is of magnitude $\sqrt{\dfrac23}$, is:

Accepted Answer

Let $\vec d=\vec b+\lambda\vec c=(1+\lambda)\hat i+(2+\lambda)\hat j-(1+2\lambda)\hat k$. Projection of $\vec d$ on $\vec a$ is $\dfrac{\vec d\cdot\vec a}{|\vec a|}$, with $|\vec a|=\sqrt6$:

$$\vec d\cdot\vec a = 2(1+\lambda)-(2+\lambda)-(1+2\lambda) = -1-\lambda$$

$$\left(\frac{-(1+\lambda)}{\sqrt6}\right)^2=\frac23 \implies \frac{(1+\lambda)^2}6=\frac23 \implies (1+\lambda)^2=4 \implies \lambda^2+2\lambda-3=0 \implies \lambda=1,-3$$

For $\lambda=1$: $\vec b+\lambda\vec c=2\hat i+3\hat j-3\hat k$ (option B). For $\lambda=-3$: $\vec b+\lambda\vec c=-2\hat i-\hat j+5\hat k$ (not among the options).

Solutions · 1