Question

B -B (A+B) (B+A) (A-B) B FIGURE 1.3 FIGURE 1.4 (1) Addition of two vectors. Place the tail of B at the head of A; the sum, A+B, is the vector from the tail of A to the head of B (Fig. 1.3). (This rule generalizes the obvious procedure for combining two displacements. Addition is commutative: A+B=B+A; 3 miles east followed by 4 miles north gets you to the same place as 4 miles north followed by 3 miles east. Addition is also associative: (A+B) + C = A +(B+C). To subtract a vector, add its opposite (Fig. 1.4): A - B = A +(-B). (ii) Multiplication by a scalar. Multiplication of a vector by a positive scalar a multiplies the magnitude but leaves the direction unchanged (Fig. 1.5). (If a is negative, the direction is reversed.) Scalar multiplication is distributive: a(A + B) = aA +aB. (iii) Dot product of two vectors. The dot product of two vectors is defined by A B = AB cose, where is the angle they form when placed tail-to-tail (Fig. 1.6). Note that A.B is itself a scalar (hence the alternative name scalar product). The dot product is commutative, AB=B.A, and distributive, A. (B+C) = A B +AC. (1.2) Geometrically, A B is the product of A times the projection of B along A (or the product of B times the projection of A along B). If the two vectors are parallel, then A. B = AB. In particular, for any vector A, AA= A. (1.3) If A and B are perpendicular, then A. B = 0. (1.1)

Full answers and working out please.

Answer 1

1. Dot product is distributive: To prove: A. (B+C) = A.B + A.C we have A B = AB cose A. (B+C) - A (B+C) cose A B cose + AC cose A.(B+C) = A.B + A-C Cross product is distributive: To prove: AX (B+C) = (AXB)+ (AXC) we have AXB = AB sine î. AX (B+C) = A.(B+C) sino = AB sino ñ + ACsino - Ax (B+C) = (AXB) + (AXC) a are The three vectors are coplanar if they linearly dependent. Since the dimension of the plane is two. so the three vectors should be linearly dependent then only dimension will be two or less.

(6 For general case (ie) for n vectors to be coplanar it among them no more than two linearly in dependent vectors. Then only dimension will be two or less.