Question

Let a = `hati + 2hatj + hatk`, b = `hati - hatj + hatk`, c = `hati + hatj - hatk`. A vector coplanar to a and b has a projection along with c of magnitude `1/sqrt(3)`, then the vector is ______.

Options

• `4hati - hatj + 4hatk`

• `4hati + hatj - 4hatk`

• `2hati + hatj + hatk`

• None of these

Answer 1

Let a = `hati + 2hatj + hatk`, b = `hati - hatj + hatk`, c = `hati + hatj - hatk`. A vector coplanar to a and b has a projection along with c of magnitude `1/sqrt(3)`, then the vector is `underlinebb(4hati - hatj + 4hatk)`.

Explanation:

Let vector r be coplanar to a and b.

∴ r = a + tb

Given, a = `hati + 2hatj + hatk`, b = `hati - hatj + hatk` and c = `hati + hatj - hatk`

`\implies` r = `(hati + 2hatj + hatk) + t(hati - hatj + hatk)`

= `(1 + t)hati + (2 - t)hatj + (1 + t)hatk`

The projection of r along with c = `1/sqrt(3)`  ...(given)

`\implies (r.c)/|c| = 1/sqrt(3)`

`\implies` (2 – t) = ± 1

`\implies` t = 1 or 3

When t = 1, we have r = `2hati + hatj + 2hatk`

When t = 3, we have r = `4hati - hatj + 4hatk`