Vector Arithmetic

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number), which changes the magnitude of the vector and possibly its direction, depending on the sign of the scalar.

• Definition: If \( \vec{A} \) is a vector and c is a scalar, then the product \( \vec{B} = c\vec{A} \).

• Magnitude: \( B = |c|A \), where \( A \) is the magnitude of \( \vec{A} \).

• Direction: The direction of \( \vec{B} \) is the same as \( \vec{A} \) if c is positive, and reversed if c is negative.

• Components: If \( \vec{A} = (A_x, A_y, A_z) \), then \( \vec{B} = (cA_x, cA_y, cA_z) \).

• Units: If c has units, \( \vec{B} \) may represent a different physical quantity than \( \vec{A} \). If c is dimensionless, \( \vec{B} \) has the same units as \( \vec{A} \).

Common Cases:

• \( |c| > 1 \): Stretches the vector (increases magnitude).

• \( |c| < 1 \): Shrinks the vector (decreases magnitude).

• \( c = -1 \): Reverses the direction.

• \( c = 0 \): Results in the zero vector (no direction).

Vector Addition

Vector addition combines two or more vectors to produce a resultant vector. This operation is fundamental in physics for combining forces, velocities, and other vector quantities.

• Definition: The sum of vectors \( \vec{A} \) and \( \vec{B} \) is \( \vec{R} = \vec{A} + \vec{B} \).

• Graphical Method (Tip-to-Tail):

  • Draw \( \vec{A} \).

  • Draw \( \vec{B} \) starting from the tip of \( \vec{A} \).

  • The resultant \( \vec{R} \) is drawn from the tail of \( \vec{A} \) to the tip of \( \vec{B} \).

• Commutativity: \( \vec{A} + \vec{B} = \vec{B} + \vec{A} \).

• Adding Multiple Vectors: Repeat the tip-to-tail method for each additional vector. The resultant is from the start of the first to the end of the last vector.

Addition of Parallel and Antiparallel Vectors

• If vectors are parallel, their magnitudes add directly.

• If vectors are antiparallel, their magnitudes subtract (direction is determined by the larger magnitude).

Vector Subtraction

Vector subtraction finds the difference between two vectors. Subtracting \( \vec{B} \) from \( \vec{A} \) is equivalent to adding \( -\vec{B} \) to \( \vec{A} \).

• Definition: \( \vec{R} = \vec{A} - \vec{B} \).

• Graphical Method:

  • Draw \( \vec{A} \).

  • Draw \( -\vec{B} \) (reverse direction of \( \vec{B} \)) starting from the tip of \( \vec{A} \).

  • Resultant \( \vec{R} \) is from the tail of \( \vec{A} \) to the tip of \( -\vec{B} \).

• Non-Commutativity: \( \vec{A} - \vec{B} \neq \vec{B} - \vec{A} \); in fact, \( \vec{A} - \vec{B} = - (\vec{B} - \vec{A}) \).

Methods for Calculating Resultant Vectors

Triangle (Geometric/Trigonometric) Method

This method uses geometric and trigonometric relationships to solve for unknown sides or angles in vector addition, especially useful for two vectors or three vectors summing to zero.

• Redraw the vectors as a triangle, labeling all known sides and angles.

• Apply the Pythagorean theorem for right triangles:

• Use SOH CAH TOA for right triangles, or the Law of Sines and Cosines for general triangles:

Component Method

The component method breaks each vector into its x, y (and z, if needed) components, adds the components, and reconstructs the resultant vector.

• Find components:

• Add components:

• Resultant magnitude and direction:

Review Quiz

1. If vector \( \vec{V} \) has a trig. direction of 115°, then \( -\vec{V} \) has a trig. direction of: Answer: 295° (add 180° to reverse direction)

2. True/False: Magnitude of (A+B) = (Magnitude of A) + (Magnitude of B) for any two vectors A & B. Answer: False (only true if vectors are parallel and in the same direction)

3. If (A–B) has a trig. direction of 70°, then (B–A) has a trig. direction of: Answer: 250° (add 180° to reverse direction)