Find the direction cosines of the vector . Find the direction cosines of the vector joining the points A ( , , – ) and B (– , – , ), directed from A to B. . Show that the vector is equally inclined to the axes OX, OY and OZ.
. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are and – respectively, in the ratio : (i) internally (ii) externally . Find the position vector of the mid point of the vector joining the points P( , , ) and Q( , , – ). .
Show that the points A, B and C with position vectors, = , i and = , respectively form the vertices of a right angled triangle. . In triangle ABC (Fig . ), which of the following is not true: (A) (B) (C) (D) .
If are two collinear vectors, then which of the following are incorrect: (A) (B) (C) the respective components of are not proportional (D) both the vectors have same direction, but different magnitudes. . Product of Two Vectors So far we have studied about addition and subtraction of vectors. An other algebraic operation which we intend to discuss regarding vectors is their product.
We may recall that product of two numbers is a number, product of two matrices is again a matrix. But in case of functions, we may multiply them in two ways, namely, multiplication of two functions pointwise and composition of two functions. Similarly, multiplication of two vectors is also defined in two ways, namely, scalar (or dot) product where the result is a scalar, and vector (or cross) product where the result is a vector. Based upon these two types of products for vectors, they have found various applications in geometry, mechanics and engineering.
In this section, we will discuss these two types of products. . . Scalar (or dot) product of two vectors Definition The scalar product of two nonzero vectors , denoted by , is Fig .
defined as where, θ is