Quantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors.

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1 Vectors summary Quantities which have only magnitude are called scalars. Quantities which have magnitude and direction are called vectors. AB is the position vector of B relative to A and is the vector which emanates from A and terminates at B and has length AB. Two vectors are equal if they have the same magnitude and direction. They will be parallel and have the same length. BA AB has the same magnitude but opposite direction to AB. Other notations: If O,,, A a,, then a OA Operations with vectors ã ã a Addition: To construct a + b draw b at the arrowhead end of a. arrowhead end of b. Join beginning of a to Vector subtraction: a b a + b Zero vector: a a Scalar multiplication: ka stretches a by factor k where k is a scalar. k > ka is in the same direction as a. k < ka is in the opposite direction as a. -D vectors in component form If the horizontal and vertical components of v are x and y respectively, v vector. +b Addition: a + b + b +b Negative vectors: a Zero vector: a a. Vector subtraction: a b b b b. x y as a column

2 Vectors between two points: A a,, B b, b AB k k Scalar multiplication: ka k Length of a vector: a a + a. These can be generalised to higher dimensions. a b. 3-D coordinate geometry If O,,, P x, y, z then OP x + y + z and x OP y. z Likewise, if A x, y, z, B x, y, z then and AB x x y y z z M midpt AB midpoint of AB x +x, y +y, z +z Operations with 3-D vectors a, b b a + b k k a scalar ka. k k OA a, OB b AB b a. Division of a line segment +b +b + AB x x + y y + z z b, a b b Suppose A, B, X lie on a straight line. Then X divides AB in ratio a : b if AX : XB a : b. a >, b > internal division a >, b < external division & AX : XB a : b. Parallelism a b if a kb for some scalar k. A, B, C are collinear if AB k BC for some scalar k.

3 Unit vectors A unit vector is a vector of length. Base unit vectors are unit vectors which can be used as linear combinations to write any other vector. The most commonly used base unit vectors in 3-D are i, j, k. The unit vector in direction of u is denoted û and is equal to u u. Scalar product Also called dot product, or inner product a, b a b a b + b +. b Angle between vectors: If θ is the angle between a and b then θ cos â ˆb. Proof. By the cosine rule, b a a + b a b cos θ. But b a b a + b b + b + b 3 a b cos θ a b a b + b + a b a b cos θ cos θ a b a b θ cos â ˆb Note that this can also be used to generalise the concept of angle to higher dimensions. Perpendicular vectors For nonzero vectors a, b, a b a b since cos 9 Parallel vectors For nonzero vectors a, b, a b a b ± a b since cos and cos 8. Vector Product c If a, b b, c c, the vector product or cross product of a and b is c 3 i j k a b a b i a b j + a a b b b k b b b a a b b a b a and a b b because a a b b a b. Note also that a b b a a a b c b b is called the scalar triple product of a, b and c. c c c 3

4 Note that although it can also be written as a b c, these parentheses are not actually necessary. Length of a b a b a b sin θ where θ is the angle between a and b. Proof. a b b + b a + a b b + a + a 3 b + b + a b + b + check this Parallel vectors a b a b a b a b cos θ a b cos θ a b sin θ since θ 8. For nonzero vectors a, b, a b a b since sin sin 8 Area of triangle The area of a triangle defined by a, b is a b since it is a b sin θ Area of parallelogram The area of a parallelogram defined by a, b is a b since it is a b Volume of a parallelepiped The volume of a parallelepiped defined by a, b, c is a b c. Proof. If θ is the angle between a and the plane defined by b, c and ϕ is the angle between a and b c then the parallelogramic base defined by b, c has area b c and the height is a sin θ and hence Volume area of base height b c a sin θ a b c cos ϕ a b c Volume of a tetrahedron The volume of a tetrahedron defined by a, b, c is 6 a b c

5 Proof. If θ is the angle between a and the plane defined by b, c and ϕ is the angle between a and b c then the triangular base defined by b, c has area b c and the height is a sin θ and hence Volume 3 area of base height 3 b c a sin θ 6 a b c cos ϕ 6 a b c Test for coplanar points If A, B, C, D have position vectors a, b, c, d, then A, B, C, D are coplanar b a c a d a because the volume of the tetrahedron defined by AB, AC, AD would be. Projection vectors The vector projection of u in direction v is u ˆvˆv Proof. Proj v u u cos θ v u u v u v where θ is the angle between u, v u v v actually this is probably the most useful form of this v u v u ˆvˆv u cos θ u ˆv is called the scalar projection of u in direction v. The vector projection of u in a direction orthogonal to v is u Proj v u.

Prepared by: M. S. KumarSwamy, TGT(Maths) Page

Prepared by: M. S. KumarSwamy, TGT(Maths) Page Prepared by: M S KumarSwamy, TGT(Maths) Page - 119 - CHAPTER 10: VECTOR ALGEBRA QUICK REVISION (Important Concepts & Formulae) MARKS WEIGHTAGE 06 marks Vector The line l to the line segment AB, then a