Mathematics

Problem

Let

a, b, c

be distinct non-negative numbers. If the vectors

a\hat{i} + a\hat{j} + c\hat{k}

\hat{i} + \hat{k}

and

c\hat{i} + c\hat{j} + b\hat{k}

lie in a plane, then

c

is a. HM of

a

and

b

b. 0 c. AM of

a

and

b

d. GM of

a

and

b

Intuitive explanation

Three vectors are coplanar when one can be written as a linear combination of the other two—equivalently, their scalar triple product equals zero. Here, you'll set up the determinant condition for coplanarity and solve for $c$ in terms of $a$ and $b$ , then use the constraint that $a$ , $b$ , $c$ are distinct to pin down the exact value. The key is recognizing that the determinant equation will give you a relationship that forces $c$ into a specific form independent of the other variables.

Intuitive explanation

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