Question

If \[ \vec a=\hat i-\hat j+2\hat k,\qquad \vec b=-\hat i+2\hat j \] and \(\vec c\) are three vectors such that \(\vec a+\vec b\) is parallel to \(\vec c\) and \[ (\vec c+\vec a)\cdot(\vec c+\vec b)=7, \] then the vector \(\vec c\) having minimum length is

• A. \(\dfrac{1}{2}\hat j+\hat k\)
• B. \(\hat j+2\hat k\)
• C. \(-2\hat j-4\hat k\)
• D. \(\dfrac{1}{3}\hat j+\dfrac{2}{3}\hat k\)

Hint:

Whenever one vector is parallel to another, write it as a scalar multiple. Substitute into the given vector equation, solve for the scalar, and then compare magnitudes if a minimum or maximum length is required.

Answer 1

The Correct Option is B

Concept: If \(\vec c\) is parallel to \(\vec a+\vec b\), then \[ \vec c=\lambda(\vec a+\vec b). \] Use the given dot product condition to determine \(\lambda\), then choose the value giving the minimum magnitude of \(\vec c\).

Step 1: Find \(\vec a+\vec b\). Given, \[ \vec a=\hat i-\hat j+2\hat k, \qquad \vec b=-\hat i+2\hat j. \] Therefore, \[ \vec a+\vec b = (\hat i-\hat i)+(-\hat j+2\hat j)+(2\hat k) = \hat j+2\hat k. \] Since \(\vec c\parallel(\vec a+\vec b)\), \[ \vec c=\lambda(\hat j+2\hat k). \]

Step 2: Compute \((\vec c+\vec a)\cdot(\vec c+\vec b)\). \[ \vec c+\vec a = \hat i+(\lambda-1)\hat j+(2\lambda+2)\hat k, \] \[ \vec c+\vec b = -\hat i+(\lambda+2)\hat j+2\lambda\hat k. \] Hence, \[ (\vec c+\vec a)\cdot(\vec c+\vec b) = -1+(\lambda-1)(\lambda+2)+(2\lambda+2)(2\lambda). \] Using the given condition, \[ -1+(\lambda^2+\lambda-2)+(4\lambda^2+4\lambda)=7. \] \[ 5\lambda^2+5\lambda-3=7. \] \[ 5\lambda^2+5\lambda-10=0. \] \[ \lambda^2+\lambda-2=0. \] \[ (\lambda-1)(\lambda+2)=0. \] Thus, \[ \lambda=1 \quad\text{or}\quad \lambda=-2. \]

Step 3: Find the vector of minimum length. For \(\lambda=1\), \[ \vec c=\hat j+2\hat k, \] \[ |\vec c| = \sqrt{1^2+2^2} = \sqrt5. \] For \(\lambda=-2\), \[ \vec c=-2\hat j-4\hat k, \] \[ |\vec c| = \sqrt{(-2)^2+(-4)^2} = 2\sqrt5. \] Since \[ \sqrt5\lt 2\sqrt5, \] the minimum length occurs when \[ \vec c=\hat j+2\hat k. \] \[ \boxed{\vec c=\hat j+2\hat k} \] \[ \boxed{\text{Answer = (B)}} \]