Comsats Institute of Information Technology (CIIT), Islamabad Calculus-III Lecture No 4 by Dr. Umer Farooq

Outlines of Lecture 03 Dot Product Direction Angles and Direction Cosines Decomposition of a vector into Orthogonal Vectors

Dot Product and Projections Example 4.1 Let and be the angles shown in the accompanying figure. Find the direction cosines of . Solution:

Dot Product and Projections

Dot Product and Projections Example 4.2 The accompanying figure shows a cube. (a)Find the angle between the vectors and to the nearest degree. (b)Make a conjecture about the angle between the vectors and ,and confirm your conjecture by computing the angle. Solution:

Dot Product and Projections

Dot Product and Projections

Dot Product and Projections Work Done The work W done on the object by a constant force of magnitude acting in the direction of motion over a distance d to be force *distance Furthermore, if we assume that the object moves along A line from to , then , so that work can be expressed entirely in vector form as

Dot Product and Projections Work Done In the case where a constant force is not in the direction of motion, but rather makes an angle with the displacement vector, then we define the work done by to be

Dot Product and Projections Example 4.3 A wagon is pulled horizontally by exciting a constant force of 10lb at an angle of with the horizontal. How much work is done in moving the wagon 50ft? A force lb is applied to a point that moves on a line from to . If distance is measured in feet, how much work is done. Solution:

Dot Product and Projections

Dot Product and Projections

Cross Products Determinants

Cross Products Determinants

Cross Products Properties of Determinants: If two rows in the array of a determinant are the same, then the value of the determinant is 0. Interchanging two rows in the array of a determinant multiplies its value by -1.

Cross Products Definition: If and are vectors in 3-space then the cross product is the vector defined by

Cross Products

Cross Products

Cross Products Algebraic Properties of Cross Product: If and are vectors in 3-space then the following properties hold

Cross Products Note: (a)The cross product is defined only for vectors in 3-space, whereas the dot product is defined for vectors in 2-space and 3-space (b) The cross product of two vectors is a vector, whereas the dot product of two vectors is a scalar

Cross Products Geometric Properties of the Cross Product: If and are vectors in 3-space, then

Cross Products Example 4.4 Find a vector that is orthogonal to both of the vectors and . Solution:

Cross Products Geometric Properties of the Cross Product: Let and be non zero vectors in 3-space, and let be angle between them

Cross Products

Cross Products

Cross Products Example 4.5 Find the area of the triangle that is determined by the points , and . Solution:

Cross Products

Cross Products Scalar Triple Product: If , and are three vectors in 3-space then the scalar triple product

Cross Products

Cross Products Geometric Properties of Scalar Triple Product: Let , and are non zero vectors in 3-space then

Cross Products Example 4.5 Find two unit vectors that are normal to the plane determined by the points , and Solution:

Cross Products Example 4.5 Find two unit vectors that are parallel to the yz-plane and are orthogonal to the vector Solution:

Cross Products

Vectors Have a Nice Day Thank You