Question

Let `veca = a_1hati + a_2hatj + a_3hatk  a_i > 0`, i = 1, 2, 3 be a vector which makes equal angles with the coordinates axes OX, OY and OZ. Also, let the projection of `veca` on the vector `3hati + 4hatj` be 7. Let `vecb` a vector obtained by rotating `veca` with 90°. If `veca, vecb` and x-axis are coplanar, then projection of a vector `vecb` on `3hati + 4hatj` is equal to ______.

विकल्प

• `sqrt(7)`

• `sqrt(2)`

• 2

• 7

Answer 1

Let `veca = a_1hati + a_2hatj + a_3hatk  a_i > 0`, i = 1, 2, 3 be a vector which makes equal angles with the coordinates axes OX, OY and OZ. Also, let the projection of `veca` on the vector `3hati + 4hatj` be 7. Let `vecb` a vector obtained by rotating `veca` with 90°. If `veca, vecb` and x-axis are coplanar, then projection of a vector `vecb` on `3hati + 4hatj` is equal to `underlinebb(sqrt(2))`.

Explanation:

Given vector is 

`veca = a_1hati + a_2hatj + a_3hatk`

When the angle is equal then cos α = `1/sqrt(3)`

`veca = r(1/sqrt(3)hati + 1/sqrt(3)hatj + 1/sqrt(3)hatk) = r/sqrt(3)(hati + hatj + hatk)`

Now projection of `veca` on `vecc` = 7

Here `vecc = 3hati + 4hatj \implies (veca.vecc)/|vecc|` = 7

`r/sqrt(3) ((hati + hatj + hatk).(3hati + 4hatj))/5 = 7 \implies` r = `5sqrt(3)`

Put the value of r in the value of `veca`

`veca = 5(hati + hatj + hatk)`

As `veca, vecb` and x-axis are coplanar then

`vecb = 5λ(hati + hatj + hatk) + μ(hati)`

Take, `veca.vecb = cos90^circ |veca||vecb|`

`\implies` 25λ(3) + 5μ = 0

`\implies` 15λ + μ = 0

`\implies` μ = –15λ

`vecb = 5λ(-2hati + hatj + hatk)`

Take modulus both side

`|vecb| = 5sqrt(3) \implies` λ = `± 1/sqrt(2)`

`vecb = ± 5/sqrt(2)(-2hati + hatj + hatk)`

Projection of `vecb` on `3hati + 4hatj` is 

`(vecb.(3hati + 4hatj))/5 = ± 5/sqrt(2)((-6 + 4)/5) = ± sqrt(2)`