Let a unit vector which makes an angle of 60° with 2 î+2 ĵ-k̂ and an angle of 45° with î-k̂ be C⃗. Then C⃗+(-1/2 î+1/(3 √2) ĵ-√2/3 k̂) is:

Solution

Maths sir · 4 steps

Each angle gives one dot-product equation and the unit length gives a third: find C, then add.

1. Angles to equations. Let C⃗ = xî+yĵ+zk̂ with |C⃗| = 1. The magnitudes are |2î+2ĵ-k̂| = 3 and |î-k̂| = √2. So the dot products are 1 × 3 × cos 60° = 3/2 and 1 × √2 × cos 45° = 1.

   2x+2y-z=3/2, x-z=1, x²+y²+z²=1

2. Reduce to x. The second equation gives z = x-1. Putting this in the first gives x+2y=1/2, so y=1/4-x/2. The unit-length condition then becomes 36x²-36x+1=0, so x=1/2±√2/3.
3. Test both roots. Take x=1/2+√2/3. Then y=-√2/6=-1/3√2 and z=-1/2+√2/3. Adding the given vector: the î part is √2/3, the ĵ part is 0, and the k̂ part is -1/2. The other root gives -√2/3î+√2/3ĵ-(1/2+2√2/3)k̂, which matches no option.
4. So. The sum is √2/3î-1/2k̂, which is option (c).

The idea to keep. An unknown unit vector needs three equations: one dot product for each angle, plus x²+y²+z²=1. The quadratic gives two vectors, so check both against the options.

Answer from the NTA final answer key. Solution by PrepWiser.