If , where , and are all non-zero vectors, show that bisects the angle between and .

You may consider the scalar product of and , and and .

Let be the angle between and .

Let be the angle between and .

(Our goal is to show that .)

— (1)

— (2)

Using (1), we have

Using (2), we have

Since , then the parallelogram with sides determined by vectors and is a rhombus and corresponds to its diagonal. But a diagonal of a rhombus bisects its angle: the obtained two triangles are congruent by SSS. Clearly the same argument gives a more general statement: the sum of two vectors of equal length bisects the angle between them.

9758 A-level A-levels A'levels application H2 Mathematics How To Mathematics Pure Mathematics Scalar Product Vectors

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