Q.32 Let OR be the vector that is perpendicular to the vectors OP = 2i − 3j + k and OQ = −2i + j + k.
If the length of the vector OR is α√3, then α is ______.
- 3
- 4
- 5
- 6
Find α Where |OR| = α√3
To find a vector OR perpendicular to both vectors OP and OQ, we use the cross product:
OR = OP × OQ.
Given Vectors
- OP = 2i – 3j + k
- OQ = -2i + j + k
Cross Product Calculation
Using the determinant formula:
OR = |i j k|
|2 -3 1|
|-2 1 1|
Step 1: Compute Components
- i-component: (-3)(1) – (1)(1) = -3 – 1 = -4
- j-component: -[(2)(1) – (1)(-2)] = -(2 + 2) = -4
- k-component: (2)(1) – (-3)(-2) = 2 – 6 = -4
Therefore, OR = -4i – 4j – 4k
Step 2: Magnitude of OR
|OR| = √[(-4)² + (-4)² + (-4)²] = √(16 + 16 + 16) = √48 = 4√3
Given |OR| = α√3, we have α = 4
Option Analysis
| Option | Why Incorrect/Correct |
|---|---|
| 3 | Magnitude too small; cross product yields 4√3 > 3√3. |
| 4 | Matches calculation: |OR| = 4√3. Correct! |
| 5 | Would correspond to different vector components (~5√3). Does not match. |
| 6 | Too large; does not match cross product magnitude. |
Conclusion
The perpendicular vector OR has magnitude exactly 4√3, confirming α = 4.
