Let a⃗=2 î-3 ĵ+4 k̂, b⃗=3 î+4 ĵ-5 k̂ and a vector c⃗ be such that a⃗ ×(b⃗+c⃗)+b⃗ × c⃗=î+8 ĵ+13 k̂. If a⃗ · c⃗=13, then (24-b⃗ · c⃗) is equal to _______.

Solution

Maths sir · 5 steps

Expand the cross products and regroup: everything with c⃗ becomes (a⃗+b⃗)×c⃗.

1. Regroup the terms. Open the bracket: a⃗×b⃗ + a⃗×c⃗ + b⃗×c⃗. The two terms with c⃗ combine into (a⃗+b⃗)×c⃗. Move a⃗×b⃗ to the right side.

   (a⃗+b⃗)×c⃗ = (î+8ĵ+13k̂) - a⃗×b⃗

2. Known vectors. a⃗×b⃗ = -î+22ĵ+17k̂, so the right side is 2î-14ĵ-4k̂. Also a⃗+b⃗ = 5î+ĵ-k̂.
3. Compare components. Take c⃗ = xî+yĵ+zk̂. Then (a⃗+b⃗)×c⃗ = (y+z)î - (x+5z)ĵ + (5y-x)k̂, which gives y+z=2 and x=5y+4. The third equation x+5z=14 follows from these two, so the cross product alone cannot fix c⃗.
4. Use the dot product. Put x=5y+4 and z=2-y into a⃗·c⃗ = 2x-3y+4z = 13. This gives 3y+16=13, so y=-1, x=-1 and z=3.
5. So. b⃗·c⃗ = -3-4-15 = -22, and 24-(-22) = 46.

The common slip: 2. Treating b⃗·c⃗=-22 as +22 in the last step gives 2. Subtracting a negative adds it.

The idea to keep. When c⃗ appears in several cross products, collect them into one cross product. A cross-product equation fixes c⃗ only up to a line, so the dot condition is what pins it down.

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