Question

Let \(\vec{a}=2\hat{i}-\hat{j}+2\hat{k}\) and \(\vec{b}=\hat{i}+2\hat{j}-\hat{k}\). Let a vector \(\vec{v}\) be in the plane containing \(\vec{a}\) and \(\vec{b}\). If \(\vec{v}\) is perpendicular to the vector \(3\hat{i}+2\hat{j}-\hat{k}\) and its projection on \(\vec{a}\) is 19 units, then \(\left|2\vec{v}\right|^2\) is equal to ______________.

Hint:

Write \(\vec v=\lambda\vec a+\mu\vec b\); the perpendicularity gives \(\lambda=-4\mu\), and the projection condition gives \(\vec v\cdot\vec a=57\).

Answer 1

Correct Answer: 1494

Step 1: Let \(\vec v=\lambda\vec a+\mu\vec b\) and \(\vec n=3\hat i+2\hat j-\hat k\). \(\vec a\cdot\vec n=6-2-2=2\), \(\vec b\cdot\vec n=3+4+1=8\).
Step 2: \(\vec v\cdot\vec n=0\Rightarrow 2\lambda+8\mu=0\Rightarrow\lambda=-4\mu\). So \(\vec v=\mu(\vec b-4\vec a)=\mu(-7\hat i+6\hat j-9\hat k)\).
Step 3: Projection on \(\vec a\): \(\frac{\vec v\cdot\vec a}{|\vec a|}=19\), \(|\vec a|=3\Rightarrow\vec v\cdot\vec a=57\). \(\vec v\cdot\vec a=\mu(-14-6-18)=-38\mu=57\Rightarrow\mu=-\frac32\).
Step 4: \(|\vec v|^2=\mu^2(49+36+81)=\frac94\cdot166=373.5\).
Step 5: \(|2\vec v|^2=4\times373.5=1494\).
Answer: 1494