Question

If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k", hat"i" + hat"k", "c"hat"i" + "c"hat"j" + "b"hat"k"` are coplanar, then c is ______.

Options

• A.M. of a and b

• G.M. of a and b

• Sum of a and b

• H.M. of a and b

Answer 1

If the vectors `"a"hat"i" + "a"hat"j" + "c"hat"k", hat"i" + hat"k", "c"hat"i" + "c"hat"j" + "b"hat"k"` are coplanar, then c is G.M. of a and b.

Explanation:

Since the given vectors are coplanar,

`|("a", "a", "c"),(1, 0, 1),("c", "c", "b")|` = 0

⇒ `-"ac" - "a"("b" - "c") + "c"^2` = 0

⇒ `-"ac" - "ab" + "ac" + "c"^2` = 0

⇒ c² = ab

⇒ c is the G. M. of a and b.