Information about Class 12 Maths - Vectors

Class 12 Maths - Vectors by ednexa.com

c d 2q c d b a ____________ 2c 2q b a 2q b 2c a 2q b 2c a 2 1 2 1 This means that N divides BQ internally in the ratio 2: 1 and N divides AC internally in the ratio 2 : 1. From (iv) and (v), if follows that DP and BQ trisect AC and are trisected by AC. 2. If in a tetrahedron, the two pairs of opposite edges are perpendicular, then show that the edges in the third pair is also perpendicular. Sol: Let O-ABC be a tetrahedron. Then (OA, BC), (OB, CA) and (OC, AB) are the pair of opposite edges. Take O as the origin of reference and let a, b and c be the position vectors of the vertices A, B and C respectively. Then OA a, OB b, OC c, AB b a, BC c b and CA a c. Now, suppose the pair (OA, BC) and (OB, CA) are perpendicular to each other. Then

OA. BC 0, i.e., a. (c b) 0 a. c a. b 0 ...(i) and OB. CA 0, i.e., b. (a c) 0 b. a b. c 0 a. b b. c 0 Adding (i) and (2), we get, ...(ii) a. c b. c 0 c. b c. a 0 i.e., c. (b a) 0 OC. AB 0 ∴ the third pair (OC, AB) is perpendicular. 3. Show that the line segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length of that of the third side. Sol: Let ABC be a triangle and M and N be the midpoints of the AB and AC. Then we have to show MN parallel to BC and 1 (MN) . (BC). 2 Let a, b, c, m and n be the position vectors of A, B, C, M and N respectively. Since M and N are the midpoints of AB and AC respectively,

ab a c and n 2 2 MN n m m ac ab 2 2 1 (a c a b) 2 1 (c b) 2 1 BC 2 Thus MN is non –zero scalar multiple of BC . MN is parallel to BC. seg MN is parallel to seg BC. 1 Also, | MN | | BC | 2 1 (MN) . (BC). 2 Keep on visiting www.ednexa.com for more study material. -Team Ednexa

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