Question

If two vectors P = $\hat{i}$+2m$\hat{j}$+m$\hat{k}$ & Q = 4$\hat{i}$-2$\hat{j}$+m$\hat{k}$ are perpendicular to each other, then find value of m.

• A. m = 3
• B. m = 2
• C. m = 8
• D. m = 1

Answer 1

Answer: (B)

m = 2

Answer 2

The correct answer is (B) : m =2
P . Q = 0
($\hat{i}$+2m$\hat{j}$+m$\hat{k}$).(4$\hat{i}$-2$\hat{j}$+m$\hat{k}$) = 0
4 – 4m + m2 = 0
m2– 2m – 2m + 4 = 0
m (m – 2) – 2 (m – 2) = 0
m = 2
To determine the value of m that makes vectors P and Q perpendicular to each other, we can use the dot product of the two vectors.
The dot product of two vectors A and B is given by:
A·B = AₓBₓ + AᵧBᵧ + A₂B₂
Given :
vectors P = $\hat i$ + 2m$\hat j$ + m$\hat k$ and Q = 4$\hat i$ - 2$\hat j$ + m$\hat k$, we can calculate their dot product:
P·Q = (1)(4) + (2m)(-2) + (m)(m)
Since the vectors P and Q are perpendicular, their dot product should be zero:
0 = 4 - 4m + m²
Rearranging the equation: m² - 4m + 4 = 0
This equation can be factored as:
(m - 2)(m - 2) = 0
From this, we can see that the value of m is 2.
$\therefore$ the value of m that makes vectors P and Q perpendicular to each other is m = 2.