Question

If \[ \vec a=\hat i+p\hat j-3\hat k,\qquad \vec b=2\hat i-3\hat j+q\hat k,\qquad \vec c=\hat i+2\hat j+2\hat k\;(p0) \] are three vectors such that the magnitude of projection of \(\vec a\) on \(\vec c\) is \(3\) and the magnitude of projection of \(\vec b\) on \(\vec c\) is \(2\), then the magnitude of projection of \(\vec a\) on \(\vec b\) is

• A. \(\dfrac{11}{\sqrt{38}}\)
• B. \(\dfrac{34}{\sqrt{38}}\)
• C. \(\dfrac{7}{\sqrt{38}}\)
• D. \(\dfrac{16}{\sqrt{38}}\)

Hint:

Magnitude of projection of \(\vec a\) on \(\vec b\) is \[ \boxed{\frac{|\vec a\cdot\vec b|}{|\vec b|}}. \] Always determine unknown parameters first using the given projection conditions.

Answer 1

The Correct Option is C

Step 1: Use the formula for projection. Since \[ |\vec c| =\sqrt{1^2+2^2+2^2}=3, \] the magnitude of projection of \(\vec a\) on \(\vec c\) is \[ \frac{|\vec a\cdot\vec c|}{|\vec c|}=3. \] Hence, \[ |\vec a\cdot\vec c|=9. \] Now, \[ \vec a\cdot\vec c =1+2p-6 =2p-5. \] Therefore, \[ |2p-5|=9. \] Since \(p<0\), \[ 2p-5=-9 \] giving \[ p=-2. \]

Step 2: Find \(q\). Similarly, \[ \frac{|\vec b\cdot\vec c|}{3}=2. \] Hence, \[ |\vec b\cdot\vec c|=6. \] Now, \[ \vec b\cdot\vec c =2-6+2q =2q-4. \] Thus, \[ |2q-4|=6. \] Since \(q>0\), \[ 2q-4=6, \] so \[ q=5. \]

Step 3: Compute the required projection. Now, \[ \vec a=(1,-2,-3), \qquad \vec b=(2,-3,5). \] Therefore, \[ \vec a\cdot\vec b =2+6-15 =-7. \] Also, \[ |\vec b| = \sqrt{2^2+(-3)^2+5^2} = \sqrt{38}. \] Hence, the magnitude of the projection is \[ \frac{|\,\vec a\cdot\vec b\,|}{|\vec b|} = \frac7{\sqrt{38}}. \] Thus, \[ \boxed{\frac7{\sqrt{38}}}. \] Hence, the correct option is \(\boxed{(C)}\).