Question

If the vectors \(\vec a=2\hat i+3\hat j-\hat k\) and \(\vec b=\hat i-2\hat j+4\hat k\), then \(\vec a\cdot\vec b\) is:

• A. \(-8\)
• B. \(-4\)
• C. \(6\)
• D. \(10\)

Hint:

For perpendicular vectors: \[ \vec a\cdot\vec b=0 \] Always multiply corresponding components carefully.

Answer 1

The Correct Option is B

Concept: The dot product of vectors: \[ \vec a=a_1\hat i+a_2\hat j+a_3\hat k \] and \[ \vec b=b_1\hat i+b_2\hat j+b_3\hat k \] is: \[ \vec a\cdot\vec b=a_1b_1+a_2b_2+a_3b_3 \]

Step 1: Identifying components. Given: \[ \vec a=2\hat i+3\hat j-\hat k \] \[ \vec b=\hat i-2\hat j+4\hat k \] Thus: \[ a_1=2,\quad a_2=3,\quad a_3=-1 \] \[ b_1=1,\quad b_2=-2,\quad b_3=4 \]

Step 2: Applying the dot product formula. \[ \vec a\cdot\vec b=(2)(1)+(3)(-2)+(-1)(4) \] \[ =2-6-4 \] \[ =-8 \]