vectors worksheet with answers pdf

Lemon

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03-02-2025

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This worksheet provides a comprehensive exploration of vectors, covering fundamental concepts and applications. Each problem is designed to build your understanding, progressing from basic definitions to more complex manipulations. Solutions are provided at the end to allow for self-assessment and learning.

Section 1: Basic Vector Operations

Instructions: For each problem, perform the indicated vector operations. Assume all vectors are in R².

1. Vector Addition:

• a: u = (2, 3), v = (4, -1). Find u + v.
• b: u = (-1, 5), v = (3, 2). Find u + v.
• c: u = (0, 6), v = (-2, 0). Find u + v.

2. Vector Subtraction:

• a: u = (5, 2), v = (1, 4). Find u - v.
• b: u = (-3, 1), v = (2, -5). Find u - v.
• c: u = (7, 0), v = (0, -3). Find u - v.

3. Scalar Multiplication:

• a: u = (2, 5), k = 3. Find ku.
• b: u = (-4, 1), k = -2. Find ku.
• c: u = (0, -8), k = ½. Find ku.

Section 2: Magnitude and Direction

Instructions: Calculate the magnitude and direction (angle with the positive x-axis) of each vector.

4. Magnitude and Direction:

• a: u = (3, 4)
• b: u = (-2, 2)
• c: u = (0, -5)

5. Unit Vectors:

• a: Find the unit vector in the direction of u = (6, 8).
• b: Find the unit vector in the direction of u = (-1, √3).
• c: Find the unit vector in the direction of u = (5, 0).

Section 3: Dot Product and Applications

Instructions: Perform the indicated operations.

6. Dot Product:

• a: u = (2, 3), v = (4, -1). Find u ⋅ v.
• b: u = (-1, 5), v = (3, 2). Find u ⋅ v.
• c: u = (0, 6), v = (-2, 0). Find u ⋅ v.

7. Angle Between Vectors:

Find the angle between the vectors u and v using the dot product.

• a: u = (1, 1), v = (1, -1).
• b: u = (2, 0), v = (0, 3).

Section 4: Vector Projections

Instructions: Find the projection of vector u onto vector v.

8. Vector Projection:

• a: u = (1, 2), v = (3, 4)
• b: u = (4, -1), v = (2, 1)

Answers

(Note: Answers are rounded to two decimal places where applicable. Exact answers may vary slightly depending on the method used.)

Section 1:

1. a) (6, 2) b) (2, 7) c) (-2, 6)
2. a) (4, -2) b) (-5, 6) c) (7, 3)
3. a) (6, 15) b) (8, -2) c) (0, -4)

Section 2:

4. a) Magnitude: 5, Direction: 53.13° b) Magnitude: 2.83, Direction: 135° c) Magnitude: 5, Direction: 270°
5. a) (0.6, 0.8) b) (-0.5, 0.87) c) (1, 0)

Section 3:

6. a) 5 b) 7 c) 0
7. a) 90° b) 90°

Section 4:

8. a) (1.07, 1.43) b) (1.6, 0.8)

This worksheet provides a solid foundation in vector operations. Remember to practice regularly to solidify your understanding. Further exploration of topics like cross products (for vectors in R³), vector spaces, and linear transformations will build upon this base.