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Test for Collinearity of Points

If A⃗+B⃗=P⃗ a...

Question

If $\to A$ + $\to B$ = $\to P$ and $\to A$ $\times$ $\to B$ = $\to Q$ then

A

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B

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C

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D

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Solution

The correct option is C

The resultant of A and B is P and lies in the plane of these vectors.

The cross product of A and B is Q which is perpendicular to the plane containing A and B.

Therefore, the dot product of P and Q is zero as they are perpendicular to each other.

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Similar questions

\to a, \to b, \to c

\to p, \to q, \to r

are reciprocal system of vectors, then

(\to a \times \to p) + (\to b \times \to q) + (\to c \times \to r)

\to a \times {\to b \times (\to c \times \to a) + (\to p \times \to q)} =

\to a, \to b, \to c

are non coplanar vectors and

\to p, \to q, \to r

\to p = \frac{\to b \times \to c}{[\begin{matrix} \to b & \to c & \to a \end{matrix}]}

\to q = \frac{\to c \times \to a}{[\begin{matrix} \to c & \to a & \to b \end{matrix}]}

\to r = \frac{\to a \times \to b}{[\begin{matrix} \to a & \to b & \to c \end{matrix}]}

(\to a + \to b) . \to p + (\to b + \to c) . \to q + (\to c + \to a) . \to r

\to a

\to b

be two vectors of equal magnitude

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\to p, \to q

be vectors such that

\to p = \to a - \to b

\to q = \to a + \to b

| \to p \times \to q | = 2 {γ - (\to a . \to b)^{2}}^{1 / 2}

, then the value of

γ

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are four vectors such that

sin (\to p \land \to q) = 1

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