Let a⃗=î+2 ĵ+3 k̂, b⃗=3 î+ĵ-k̂ and c⃗ be three vectors such that c⃗ is coplanar with a⃗ and b⃗. If the vector C⃗ is perpendicular to b⃗ and a⃗ · c⃗=5, then |c⃗| is equal to

Solution

Maths sir · 4 steps

Coplanar with a and b means c is a mix of them: write c = xa + yb and let the two dot conditions find x and y.

1. Set up c. Since c⃗ lies in the plane of a⃗ and b⃗, write c⃗ = xa⃗ + yb⃗. First find the dot products you will need: a⃗·a⃗ = 14, b⃗·b⃗ = 11 and a⃗·b⃗ = 3 + 2 - 3 = 2.
2. Two conditions. c⃗·b⃗ = 0 gives 2x + 11y = 0. a⃗·c⃗ = 5 gives 14x + 2y = 5.
3. Solve. From the first equation, x = -11y/2. Substituting into the second gives -77y + 2y = 5, so y = -1/15 and x = 11/30.

   c⃗ = 11/30(î+2ĵ+3k̂) - 1/15(3î+ĵ-k̂) = 1/6(î+4ĵ+7k̂)

4. So. |c⃗| = √(1+16+49)/6 = √66/6 = √(66/36) = √(11/6), which is option (a). Check: c⃗·b⃗ = (3+4-7)/6 = 0 and a⃗·c⃗ = (1+8+21)/6 = 5.

The idea to keep. Whenever a vector is coplanar with two given vectors, write it as xa⃗ + yb⃗. Each dot-product condition then gives one linear equation in x and y.

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