Physics Vectors Javid

Scalars and Vectors Temperature = Scalar Quantity is specified by a single number giving its magnitude. Velocity = Vector Quantity is specified by three numbers that give its magnitude and direction (or its components in three perpendicular directions).

Properties of Vectors Two vectors are equal if they have the same magnitude and direction.

Adding Vectors

Subtracting Vectors

Combining Vectors

Using the Tip-to-Tail Rule

Clicker Question 1 Question: Which vector shows the sum of A1 + A2 + A3 ?

Multiplication by a Scalar

Coordinate Systems and Vector Components Determining the Components of a Vector The absolute value |Ax| of the x-component Ax is the magnitude of the component vector . The sign of Ax is positive if points in the positive x-direction, negative if points in the negative x-direction. The y- and z-components, Ay and Az, are determined similarly. Knight’s Terminology: The “x-component” Ax is a scalar. The “component vector” is a vector that always points along the x axis. The “vector” is , and it can point in any direction.

Determining Components

Cartesian and Polar Coordinate Representations

Unit Vectors Example:

Working with Vectors ^ A = 100 i m B = (-200 Cos 450 i + 200 Cos 450 j ) m = (-141 i + 141 j ) m ^ ^ ^ ^ C = A + B = (100 i m) + (-141 i + 141 j ) m = (-41 i + 141 j ) m ^ ^ ^ ^ C = [Cx2 + Cy2]½ = [(-41 m)2 + (141 m)2]½ = 147 m q = Tan-1[Cy/|Cx|] = Tan-1[141/41] = 740 Note: Tan-1 Þ ATan = arc-tangent = the angle whose tangent is …

Tilted Axes Cx = C Cos q Cy = C Sin q

Arbitrary Directions

Perpendicular to a Surface

Multiplying Vectors* Given two vectors: Dot Product (Scalar Product) Cross Product (Vector Product) A·B is B times the projection of A on B, or vice versa. (determinant)

Spherical Coordinates* q R x y z f Ax Ay Az 0 £ q £ p 0 £ f £ 2p Ax= R Sin q Cos f Ay= R Sin q Sin f Az= R Cos q R = [Ax2 + Ay2 + Az2]½ = |A| = Tan-1{[Ax2 + Ay2]½/Az} f = Tan-1[Ay/Ax]

Cylindrical Coordinates* q r x y z Ax Ay Az 0 £ q £ 2p Ax= r Cos q Ay= r Sin q Az= z r = [Ax2 + Ay2]½ = Tan-1[Ay/Ax] z = Az

Summary

Summary