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Vector Algebra

Short Answer Type

221.

Prove that an angle inscribed in a semi-circle is a right angle.

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Long Answer Type

222. Using vectors, prove that if two medians of a triangle ABC be equal, then it is an isosceles triangle.

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223. Using vectors, prove that the perpendicular bisectors of the sides of a triangle are concurrent.

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Short Answer Type

224. Prove that the median to the base of isosceles triangle is perpendicular to the base.

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Long Answer Type

225. If the median of the base of a triangle is perpendicular on the base, then prove that the triangle is an isosceles.

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226. (i) Prove that cos (α + β) )= cos α cos β – sin α sin β.
(ii) Prove that cos (α – p) = cos α cos β + sin α sin β.

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227. Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.

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Short Answer Type

228. Using vectors, prove that a parallelogram whose diagonals are equal is a rectangle.

Let OACB be a parallelogram such that OC = AB

Let

OA with rightwards arrow on top space equals space straight a with rightwards arrow on top comma space space OB with rightwards arrow on top space equals space straight b with rightwards arrow on top

Now, OC = AB

rightwards double arrow space space space OC squared space equals space AB squared rightwards double arrow space space open parentheses OA with rightwards arrow on top plus AC with rightwards arrow on top close parentheses squared space equals open parentheses AO with rightwards arrow on top space plus OB with rightwards arrow on top close parentheses squared rightwards double arrow space space space open parentheses OA with rightwards arrow on top space plus space OB with rightwards arrow on top close parentheses squared space equals space open parentheses negative OA with rightwards arrow on top space plus space OB with rightwards arrow on top close parentheses squared rightwards double arrow space space left parenthesis straight a with rightwards arrow on top space plus straight b with rightwards arrow on top right parenthesis squared space equals space left parenthesis negative straight a with rightwards arrow on top space plus space straight b with rightwards arrow on top right parenthesis squared rightwards double arrow space space straight a with rightwards arrow on top squared plus space straight b with rightwards arrow on top squared space plus space 2 space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space straight a with rightwards arrow on top squared space plus space straight b with rightwards arrow on top squared space minus space 2 space straight a with rightwards arrow on top. space straight b with rightwards arrow on top rightwards double arrow space space 2 space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space minus 2 space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space space space rightwards double arrow space space space 4 space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 rightwards double arrow space space straight a with rightwards arrow on top. space straight b with rightwards arrow on top space equals space 0 with rightwards arrow on top rightwards double arrow space space space straight a with rightwards arrow on top space and space straight b with rightwards arrow on top space are space perpendicular. space space space space space space rightwards double arrow space space space angle AOB space equals space 90 degree rightwards double arrow space space space OACB space is space straight a space rectangle.

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Long Answer Type

229. Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.

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230. Show that the angle between two diagonals of a cube is

cos to the power of negative 1 end exponent space open parentheses 1 third close parentheses.

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