Question

If a vector $2\widehat i + 3\widehat j + 8\widehat k$ is perpendicular to the vector $4\widehat i - 4\widehat j + \alpha \widehat k$ then the value of $\alpha$ is
(A) $\dfrac{1}{2}$
(B) $- \dfrac{1}{2}$
(C) $1$
(D) $- 1$

Answer 1

Hint : Here we are given two vectors which are perpendicular to each other, therefore we will perform dot product taking resultant value equal to zero and find the resultant value for $\alpha$ .

Complete Step By Step Answer:
Given that the two vectors are perpendicular suggests that we have to take the dot product of equal to zero.
$A.B = 0$
Place the vector values in the above expression.
$(2i + 3j + 8k).(4i - 4j + \alpha k) = 0$
Dot product in physics is the product of two vectors. It is the sum of the products of the corresponding values in the vectors That is I, j and k. Remember the dot products as $i.i = j.j = k.k = 1$ .
$\Rightarrow 8 - 12 + 8\alpha = 0$
When you subtract a larger number from the smaller positive number, resultant value is negative.
$\Rightarrow - 4 + 8\alpha = 0$
When you move any term from one side to another, the sign of the terms also changes. Positive term changes to the negative term.
Make required term the subject –
$\Rightarrow 4 = 8\alpha$
The above equation can be re-written as –
$\Rightarrow 8\alpha = 4$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$\Rightarrow \alpha = \dfrac{4}{8}$
Common factors from the numerator and the denominator cancel each other.
$\Rightarrow \alpha = \dfrac{1}{2}$
Hence, from the given multiple choices – the option A is the correct answer.

Note :
Know the difference between the dot product and cross product and apply the concepts accordingly. Dot product follows the cumulative law whereas the cross product does not follow the cumulative law. Always remember that the cross product follows the right hand thumb rule whereas the dot product does not follow it. The result of the cross product is always the vector quantity.