Question

If `vec"a", vec"b", vec"c"` are mutually perpendicular vectors having magnitudes 1, 2, 3 respectively, then `[(vec"a" + vec"b" + vec"c", vec"b" - vec"a", vec"c")]` = ______.

Options

• 0

• 6

• 12

• 18

Answer 1

If `vec"a", vec"b", vec"c"` are mutually perpendicular vectors having magnitudes 1, 2, 3 respectively, then `[(vec"a" + vec"b" + vec"c", vec"b" - vec"a", vec"c")]` = 12.

Explanation:

`|vec"a"| = 1, |vec"b"| = 2, |vec"c"| = 3`

`[(vec"a" + vec"b" + vec"c", vec"b" - vec"a", vec"c")]`

= `(vec"a" + vec"b" + vec"c").[(vec"b" - vec"a") xx vec"c"]`

= `(vec"a" + vec"b" + vec"c").(vec"b" xx vec"c" - vec"a" xx vec"c")`

= `[(vec"a", vec"b", vec"c")] - [(vec"b", vec"a", vec"c")]`

= `2[(vec"a", vec"b", vec"c")]`

= `2 vec"a".(vec"b" xx vec"c")`

= `2|vec"a"|.|vec"b" xx vec"c"| cos 0^circ`

= `2|bar"a"|.|vec"b" xx vec"c"|`

= `2|vec"a"|.|vec"b"|.|vec"c"| sin 90^circ`

= 2 × 1 × 2 × 3

= 12