7.1 Scalars and vectors Scalar: a quantity specified by its magnitude, for example: temperature, time, mass, and density Chapter 7 Vector algebra Vector: a quantity specified by a magnitude and a direction, for example: force, momentum, and electric field

7.2 Addition and subtraction of vectors Chapter 7 Vector algebra

7.3 Multiplication by a scalar Ex: A point P divides a line segment AB in the ratio λ: μ. If the position vectors of the point A and B are and respectively, find the position vector of the point P.

Ex: The vertices of triangle ABC have position vector and relative to some origin O. Find the position vector of the centroid G of the triangle. Chapter 7 Vector algebra

7.4 Basis vector and components A basis set must (1) have as many basis vectors as the number of dimension (2) be such that the basis vectors are linearly independent In Cartesian coordinate system 7.4 Basis vectors and components

Chapter 7 Vector algebra 7.5 Magnitude of a vector 7.6 Multiplication of vectors (1) scalar product (2) vector product (1) Scalar product: The Cartesisn basis vectors are mutually orthogonal Ex: work: potential energy:

Commutative and distributive: Chapter 7 Vector algebra In terms of the components, the scalar product is given by Ex: Find the angle between the vector and

Chapter 7 Vector algebra direction cosines of vector scalar product for vectors with complex components

(1) Vector product: Chapter 7 Vector algebra Properties: Ex: The moment or torque about O is

Ex: If a solid body rotates about some axis, the velocity of any point in the body with position vector is. Chapter 7 Vector algebra For the basis vector in Cartesian coordinate:

Ex: find and the area of parallelogram. Chapter 7 Vector algebra Scalar triple product

Chapter 7 Vector algebra Useful formulas: Some basic operations:

Ex: Show that Chapter 7 Vector algebra Proof:

Equation of a line: A line passing through the fixed point A with position vector and having a direction, the position vector of a general point R on the line is Chapter 7 Vector algebra 7.7 Equations of lines, planes and sphere

Equation of a plane: Chapter 7 Vector algebra The equation of a plane containing points A, B and C with position vectors

Chapter 7 Vector algebra Ex: Find the direction of the line of intersection of the two planes x+3y-z=5 and 2x-y+4z=3. Normal vector of the planes are The direction vector of line is along the direction of Equation of a sphere with radius a:

Chapter 7 Vector algebra Ex: Find the radius of the circle that is the intersection of the plane and the sphere of radius centered on the point with position vector.

7.8 Using vectors to find distances The minimum distance from a point to a line Chapter 7 Vector algebra Ex: Find the minimum distance from the point P with coordinate (1,2,1) to the line, where

The minimum distance from a point to a plane Chapter 7 Vector algebra Ex: Find the distance from the point P with coordinate (1,2,3) to the plane that contains the point A, B and C having coordinates (0,1,0), (2,3,1) and (5,7,2).

The minimum distance from a line to a line Chapter 7 Vector algebra Ex: A line is inclined at equal angles to the x-, y- and z-axis and pass through the origin. Another line passes through the points (1,2,4) and (0,0,1). Find the minimum distance between the two lines.

The distance from a line to a plane Chapter 7 Vector algebra Ex: A line is given by, where and Find the coordinates of the point P at which the line intersects the plane x+2y+3z=6.

Chapter 7 Vector algebra 7.9 Reciprocal vectors The two sets of vectors and are called reciprocal sets if Ex: Construct the reciprocal vector of

Chapter 7 Vector algebra Define the components of a vector with respect basis vectors that are not mutually orthogonal.