Question

If \[ \vec{a}=2\hat{i}-\hat{j}+3\hat{k} \] and \[ \vec{b}=\hat{i}+4\hat{j}-2\hat{k}, \] then \(\vec{a}\cdot\vec{b}\) is equal to:

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

Hint:

For dot product: \[ \vec a\cdot\vec b = a_1b_1+a_2b_2+a_3b_3 \] Multiply corresponding components carefully and watch the signs.

Answer 1

The Correct Option is B

Concept: The dot product (scalar product) of two vectors is given by: \[ \vec{a}\cdot\vec{b} = a_1b_1+a_2b_2+a_3b_3 \] where corresponding components are multiplied and then added.

Step 1: Identify the components of the vectors. Given: \[ \vec{a}=2\hat{i}-\hat{j}+3\hat{k} \] Thus components of \(\vec a\) are: \[ (2,-1,3) \] Similarly, \[ \vec{b}=\hat{i}+4\hat{j}-2\hat{k} \] Thus components of \(\vec b\) are: \[ (1,4,-2) \]

Step 2: Apply the dot product formula. \[ \vec a\cdot\vec b = (2)(1)+(-1)(4)+(3)(-2) \] \[ = 2-4-6 \] \[ =-8 \] Hence, \[ \boxed{-8} \] Therefore the correct answer is: \[ \boxed{(B)\ -8} \]