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Question
Show that the vectors `vec"a" = 2hat"i" + 3hat"j" + 3hat"j" + 6hat"k", vec"b" = 6hat"i" + 2hat"j" - 3hat"k"` and `vec"c" = 3hat"i" - 6hat"j" + 6hat"k"` are mutually orthogonal
Solution
Given `vec"a" = 2hat"i" + 3hat"j" + 3hat"j" + 6hat"k", vec"b" = 6hat"i" + 2hat"j" - 3hat"k"` and `vec"c" = 3hat"i" - 6hat"j" + 6hat"k"`
`vec"a" * vec"b" = (2hat"i" + 3hat"j" + 6hat"k") * (6hat"i" + 2hat"j" - 3hat"k")`
= (2)(6) + (3)(2) + (6) (– 3)
= 12 + 6 – 18
= 0
∴ `vec"a"` and `vec"b"` are perpendicular.
`vec"b" * vec"c" = (6hat"i" + 2hat"j" - 3hat"k") * (3hat"i" - 6hat"j" + 6hat"k")`
= (6)(3) + (2)(– 6) + (– 3)(2)
= 18 – 12 – 6
= 0
∴ `vec"b"` and `vec"c"` are perpendicular.
`vec"c" * vec"a" = (3hat"i" - 6hat"j" + 6hat"k") * (2hat"i" + 3hat"j" + 6hat"k")`
= (3)(2) + (– 6)(3) + (2)(6)
= 6 – 18 + 12
= 0
∴ `vec"c"` and `vec"a"` are perpendicular.
Hence `vec"a", vec"b", vec"c"` are mutually perpendicular vectors.
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