Exploring Vectors: Magnitude, Direction, and Operations

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Delve into the world of vectors, understanding their properties, how to add and subtract them, and determining their components in different coordinate systems. Learn about scalar quantities, vector quantities, the tip-to-tail rule, and the role of unit vectors in calculations.

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Vectors (Knight: 3.1 to 3.4)

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 1. The absolute value |Ax| of the x-component Ax is the magnitude of the component vector . 2. The sign of Axis positive if points in the positive x-direction, negative if points in the negative x-direction. 3. The y- and z-components, Ay and Az, are determined similarly. xA x A x A Knight s Terminology: The x-component Ax is a scalar. The component vector is a vector that always points along the x axis. x A The vector is , and it can point in any direction. A

Determining Components

Cartesian and Polar Coordinate Representations

Unit Vectors i = = = = (1,0,0) unit vector in +x-direction = "i-hat" j (0,1,0) unit vector in +x-direction = "j-hat" = = (0,0,1) unit vector in +z-direction = "k-hat" k = + + = + + = ( , , ) A A A A A i A j A k A A A x y k x y z x y z Example: i j = + = (4, 2,5) 4 2 5 B k

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] = [(-41 m)2 + (141 m)2] = 147 m = Tan-1[Cy/|Cx|] = Tan-1[141/41] = 740 Note: Tan-1 ATan = arc-tangent = the angle whose tangent is