Let **a** = 2**i** + **j** + **k**, **b** = **i** − 2**j** + 3**k**, and **c** = 3**i** + **j** − 2**k** be three vectors. If **p** = **a** + λ**b** is perpendicular to **c**, then the value of λ is:

Question

Accepted Answer

We need to find λ such that **p** ⊥ **c**, which means **p** · **c** = 0.

Step 1: Express **p** in terms of λ.
**p** = **a** + λ**b**
**p** = (2**i** + **j** + **k**) + λ(**i** − 2**j** + 3**k**)
**p** = (2 + λ)**i** + (1 − 2λ)**j** + (1 + 3λ)**k**

Step 2: Use the perpendicularity condition **p** · **c** = 0.
**c** = 3**i** + **j** − 2**k**

**p** · **c** = (2 + λ)(3) + (1 − 2λ)(1) + (1 + 3λ)(−2) = 0

Step 3: Expand and simplify.
3(2 + λ) + (1 − 2λ) − 2(1 + 3λ) = 0
6 + 3λ + 1 − 2λ − 2 − 6λ = 0
5 − 5λ = 0
5λ = 5
λ = 1

Step 4: Verification.
When λ = 1, **p** = 3**i** − **j** + 4**k**
**p** · **c** = 3(3) + (−1)(1) + 4(−2) = 9 − 1 − 8 = 0 ✓

The condition of perpendicularity (dot product equals zero) is the key theorem applied here.

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