Question

Let `bar"a" = 2hat"i" + hat"j" - 2hat"k" and bar"b" = hat"i" + hat"j"`. Let `vec"c"` be a vector such that `|bar"c" - bar"a"| = 3, |(bar"a" xx bar"b") xx bar"c"|` = 3 and the angle between `vec"c" and vec"a" xx vec"b" "be" 30^circ`. Then `vec"a" * vec"c"` is equal to ______.

Options

• `1/8`

• `25/8`

• 2

• 5

Answer 1

Let `bar"a" = 2hat"i" + hat"j" - 2hat"k" and bar"b" = hat"i" + hat"j"`. Let `vec"c"` be a vector such that `|bar"c" - bar"a"| = 3, |(bar"a" xx bar"b") xx bar"c"|` = 3 and the angle between `vec"c" and vec"a" xx vec"b" "be" 30^circ`. Then `vec"a" * vec"c"` is equal to 2.

Explanation:

`bar"a" xx bar"b" = |(hat"i", hat"j", hat"k"),(2,1,-2),(1,1,0)|`

`= hat"i"(0 + 2) - hat"j"(0 + 2) + hat"k"(2 - 1)`

`= 2hat"i" - 2hat"j" + hat"k"`

Given that,

`|(bar"a" xx bar"b") xx bar"c"|` = 3

`=> |bar"a" xx bar"b"| |bar"c"| sin 30^circ = 3`

`=> (sqrt(2^2 + 2^2 + 1^2))(|bar"c"|) xx 1/2 = 3`

`=> |vec "c"| = 2`

Also, `|bar"c" - bar"a"|` = 3

`=> |bar"c"|^2 + |bar"a"|^2 - 2(bar"a" * bar"c")` = 9

`=> bar"a" * bar"c" = (|bar"c"|^2 + |bar"a"|^2 - 9)/2 = (4 + 9 - 9)/2` = 2