smc-1211-50-(Semi)circle

Vectors, EXT1 V1 2022 SPEC1 6b

is a semicircle of radius with equation . is a point on the semicircle , as shown below.

Express the vectors and in terms of , , , and , where is a unit vector in the direction of the positive -axis and is a unit vector in the direction of the positive -axis. (1 mark)

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Hence, using the vector scalar (dot) product, determine whether is perpendicular to . (3 marks)

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Three different points and are chosen on a circle centred at .

Let and . Let and let be the point such that , as shown in the diagram.

Show that and are perpendicular. (3 marks)

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In the diagram below, is a diameter of the circle with centre .

is a point on the circumference.

Using the properties of vectors , and , show that is a right angle. (2 marks)

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In the diagram above, is a diameter of the circle with centre .

is a point on the circumference of the circle.

If and , express in terms of and . (2 marks)

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