Question

If `overline"u"` and `overline"v"` are unit vectors and θ is the acute angle between them, then `2overline"u" xx 3overline"v"` is a unit vector for ______

Options

• no value of θ

• exactly one value of θ

• exactly two values of θ

• more than two values of θ

Answer 1

If `overline"u"` and `overline"v"` are unit vectors and θ is the acute angle between them, then `2overline"u" xx 3overline"v"` is a unit vector for exactly one value of θ.

Explanation:

Since, `|overline"u"| = |overline"v"| = 1` and θ is the acute angle between `overline"u"` and `overline"v"`.

∴ `|overline"u" xx overline"v"|` = sin θ .................(i)

Now, `2overline"u" xx 3overline"v"` will be a unit vector, if `|2overline"u" xx 3overline"v"| = 1`

⇒ `6|overline"u" xx overline"v"| = 1`

⇒ 6 sin θ = 1 ............[From (i)]

⇒ sin θ = `1/6`

As θ is an acute angle. So, there is only one value of θ for which `2overline"u" xx 3overline"v"` is a unit vector.