(1) Represent graphically the displacement of

(i) 45 cm 30° north of east.

(ii) 80 km 60° south of west Solution

(2) Prove that the relation R defined on the set V of all vectors by a vector R b vector if a vector = b vector is an equivalence relation on V. Solution

(3) Let a vector and b vector be the position vectors of the points A and B. Prove that the position vectors of the points which trisects the line segment AB are (a vector + 2b vector)/3 and (b vector + 2a vector)/3. Solution

(4) If D and E are the midpoints of the sides AB and AC of a triangle ABC, prove that BE vector + DC vector = (3/2) BC vector. Solution

(5) Prove that the line segment joining the midpoints of two sides of a triangle is parallel to the third side whose length is half of the length of the third side. Solution

(6) Prove that the line segments joining the midpoints of the adjacent sides of a quadrilateral form a parallelogram. Solution

(7) If a vector and b vector represent a side and a diagonal of a parallelogram, find the other sides and the other diagonal. Solution

(8) If PO vector + OQ vector = QO vector +OR vector, prove that the points P, Q, R are collinear. Solution

(9) If D is the midpoint of the side BC of a triangle ABC, prove that AB vector + AC vector = 2AD vector. Solution

(10) If G is the centroid of a triangle ABC, prove that GA vector + GB vector + GC vector = 0. Solution

(11) Let A, B, and C be the vertices of a triangle. Let D,E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that AD vector + BE vector + CF vector = 0. Solution

(12) If ABCD is a quadrilateral and E and F are the midpoints of AC and BD respectively, then prove that AB vector + AD vector + CB vector + CD vector = 4EF vector. Solution

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