Year 11 Maths: Find the Magnitude of a Vector Scalar Multiple

Introduction to Vectors

Hello, Year 11! Today, we’re going to learn about vectors, specifically how to find the magnitude of a vector when it’s multiplied by a scalar.

What is a Vector?

A vector is a quantity that has both direction and magnitude. For example, if you think about a car moving north at 60 km/h, both the speed (60 km/h) and the direction (north) together describe the vector.

What is a Scalar?

A scalar is just a number that can change the size of a vector without altering its direction. For example, if you multiply the car’s speed by 2, it would then be moving at 120 km/h in the same direction.

Finding the Magnitude of a Vector Scalar Multiple

Step-by-Step Explanation:

1. Understand the Vector:A vector is often written in the form $$\mathbf{v} = (x, y)$$ where (x) and (y) are the components in the horizontal and vertical directions, respectively.
2. Scalar Multiplication:If we multiply a vector by a scalar (k), we write this as:$$k \cdot \mathbf{v} = (k \cdot x, k \cdot y)$$This means you multiply both components of the vector by (k).
3. Magnitude of a Vector:The magnitude (or length) of a vector $$\mathbf{v}$$ is found using the formula:$$|\mathbf{v}| = \sqrt{x^2 + y^2}$$
4. Finding the Magnitude of a Scalar Multiple:If you have a scalar multiple of a vector, say $$k \cdot \mathbf{v}$$, the magnitude is given by:$$|k \cdot \mathbf{v}| = |k| \cdot |\mathbf{v}|$$This means you first find the magnitude of the original vector and then multiply it by the absolute value of the scalar.

Example:

1. Suppose we have a vector $$\mathbf{v} = (3, 4)$$.
   • The magnitude of $$\mathbf{v}$$ is:
   $$|\mathbf{v}| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$$
2. Now, if we multiply this vector by a scalar (k = 2):
   • The new vector is:
   $$k \cdot \mathbf{v} = 2 \cdot (3, 4) = (6, 8)$$
   • The magnitude of this new vector is:
   $$|2 \cdot \mathbf{v}| = 2 \cdot |\mathbf{v}| = 2 \cdot 5 = 10$$

Key Rules to Remember:

• Magnitude Formula: $$|\mathbf{v}| = \sqrt{x^2 + y^2}$$
• Scalar Multiplication: Multiply each component of the vector by the scalar.
• Magnitude of Scalar Multiple: $$|k \cdot \mathbf{v}| = |k| \cdot |\mathbf{v}|$$

Tips and Tricks:

• Always pay attention to the sign of the scalar. The magnitude is always positive, so use the absolute value.
• When finding the magnitude, remember to square the components first before adding.
• Practice with different vectors and scalars to build your confidence!

Questions

Easy Level Questions

1. Find the magnitude of $$\mathbf{v} = (1, 2)$$.
2. Find the magnitude of $$\mathbf{v} = (3, 4)$$.
3. What is the magnitude of $$2 \cdot \mathbf{v}$$ if $$\mathbf{v} = (0, 5)$$?
4. Find the magnitude of $$\mathbf{v} = (6, 8)$$.
5. What is the magnitude of $$-1 \cdot \mathbf{v}$$ if $$\mathbf{v} = (3, 4)$$?
6. Find the magnitude of $$\mathbf{v} = (5, 12)$$.
7. What is the magnitude of $$\mathbf{v} = (0, 0)$$?
8. Find the magnitude of $$3 \cdot \mathbf{v}$$ if $$\mathbf{v} = (1, 1)$$.
9. Find the magnitude of $$\mathbf{v} = (2, 2)$$.
10. What is the magnitude of $$4 \cdot \mathbf{v}$$ if $$\mathbf{v} = (-3, -4)$$?
11. Find the magnitude of $$\mathbf{v} = (7, 24)$$.
12. Find the magnitude of $$-3 \cdot \mathbf{v}$$ if $$\mathbf{v} = (1, 3)$$.
13. What is the magnitude of $$\mathbf{v} = (8, 15)$$?
14. Find the magnitude of $$\mathbf{v} = (4, 3)$$.
15. What is the magnitude of $$\mathbf{v} = (-2, -6)$$?
16. Find the magnitude of $$\mathbf{v} = (10, 0)$$.
17. What is the magnitude of $$3 \cdot \mathbf{v}$$ if $$\mathbf{v} = (1, 2)$$?
18. Find the magnitude of $$\mathbf{v} = (0, -9)$$.
19. What is the magnitude of $$5 \cdot \mathbf{v}$$ if $$\mathbf{v} = (2, 6)$$?
20. Find the magnitude of $$\mathbf{v} = (9, 12)$$.

Medium Level Questions

1. Find the magnitude of $$\mathbf{v} = (2, 3)$$, and then $$2 \cdot \mathbf{v}$$.
2. Calculate the magnitude of $$\mathbf{v} = (-4, 3)$$.
3. What is the magnitude of $$k \cdot \mathbf{v}$$ if $$\mathbf{v} = (1, 1)$$ and $$k = -3$$?
4. Find the magnitude of $$\mathbf{v} = (6, 8)$$ and then find the magnitude of $$3 \cdot \mathbf{v}$$.
5. What is the magnitude of $$\mathbf{v} = (5, -12)$$?
6. Find the magnitude of $$\mathbf{v} = (9, -40)$$.
7. Calculate the magnitude of $$2 \cdot \mathbf{v}$$ if $$\mathbf{v} = (-1, -1)$$.
8. What is the magnitude of $$\mathbf{v} = (-3, 4)$$?
9. Find the magnitude of $$\mathbf{v} = (7, 1)$$.
10. What is the magnitude of $$\mathbf{v} = (0, 8)$$?
11. Find the magnitude of $$5 \cdot \mathbf{v}$$ if $$\mathbf{v} = (-2, 2)$$.
12. What is the magnitude of $$\mathbf{v} = (3, 4)$$ and $$4 \cdot \mathbf{v}$$?
13. Find the magnitude of $$\mathbf{v} = (1, -1)$$.
14. Calculate the magnitude of $$3 \cdot \mathbf{v}$$ if $$\mathbf{v} = (5, 2)$$.
15. What is the magnitude of $$\mathbf{v} = (-7, -24)$$?
16. Find the magnitude of $$\mathbf{v} = (2, -2)$$.
17. What is the magnitude of $$\mathbf{v} = (10, 10)$$?
18. Calculate the magnitude of $$k \cdot \mathbf{v}$$ if $$\mathbf{v} = (3, 4)$$ and $$k = 2$$.
19. What is the magnitude of $$\mathbf{v} = (1, 0)$$?
20. Find the magnitude of $$4 \cdot \mathbf{v}$$ if $$\mathbf{v} = (0, 6)$$.

Hard Level Questions

1. Calculate the magnitude of $$\mathbf{v} = (3, 4)$$ and then find the magnitude of $$-2 \cdot \mathbf{v}$$.
2. What is the magnitude of $$\mathbf{v} = (8, 15)$$ and $$k \cdot \mathbf{v}$$ if $$k = 3$$?
3. Find the magnitude of $$\mathbf{v} = (5, -12)$$ and $$-3 \cdot \mathbf{v}$$.
4. What is the magnitude of $$2 \cdot \mathbf{v}$$ if $$\mathbf{v} = (1, 1)$$ and then find the magnitude of $$3 \cdot \mathbf{v}$$.
5. Calculate the magnitude of $$\mathbf{v} = (7, -24)$$ and then find the magnitude of $$-1 \cdot \mathbf{v}$$.
6. What is the magnitude of $$\mathbf{v} = (10, 0)$$ and $$k \cdot \mathbf{v}$$ if $$k = -4$$?
7. Find the magnitude of $$\mathbf{v} = (0, -9)$$ and $$-5 \cdot \mathbf{v}$$.
8. What is the magnitude of $$\mathbf{v} = (1, 2)$$ and $$k \cdot \mathbf{v}$$ if $$k = 0$$?
9. Calculate the magnitude of $$\mathbf{v} = (-6, 8)$$ and $$2 \cdot \mathbf{v}$$.
10. What is the magnitude of $$\mathbf{v} = (4, -3)$$ and $$3 \cdot \mathbf{v}$$?
11. Find the magnitude of $$\mathbf{v} = (-5, -12)$$ and $$4 \cdot \mathbf{v}$$.
12. What is the magnitude of $$\mathbf{v} = (7, 24)$$ and $$-3 \cdot \mathbf{v}$$?
13. Calculate the magnitude of $$\mathbf{v} = (0, 8)$$ and $$k \cdot \mathbf{v}$$ if $$k = -2$$.
14. What is the magnitude of $$\mathbf{v} = (10, 10)$$ and $$-5 \cdot \mathbf{v}$$?
15. Find the magnitude of $$\mathbf{v}$$(2, 2)$$ and $$4 \cdot \mathbf{v}$$.
16. Calculate the magnitude of $$3 \cdot \mathbf{v}$$ if $$\mathbf{v} = (1, -1)$$ and $$k = -4$$.
17. What is the magnitude of $$k \cdot \mathbf{v}$$ if $$\mathbf{v} = (-3, 4)$$ and $$k = 2$$?
18. Find the magnitude of $$\mathbf{v} = (1, 1)$$ and $$-3 \cdot \mathbf{v}$$.
19. What is the magnitude of $$\mathbf{v} = (8, 15)$$ and $$k \cdot \mathbf{v}$$ if $$k = -1$$?
20. Calculate the magnitude of $$\mathbf{v} = (12, -16)$$ and $$2 \cdot \mathbf{v}$$.

Answers

Easy Level Answers

1. $$|\mathbf{v}| = \sqrt{1^2 + 2^2} = \sqrt{5}$$
2. $$|\mathbf{v}| = \sqrt{3^2 + 4^2} = 5$$
3. $$|2 \cdot \mathbf{v}| = 2 \cdot 5 = 10$$
4. $$|\mathbf{v}| = \sqrt{6^2 + 8^2} = 10$$
5. $$|-1 \cdot \mathbf{v}| = 5$$
6. $$|\mathbf{v}| = \sqrt{5^2 + 12^2} = 13$$
7. $$|\mathbf{v}| = 0$$
8. $$|3 \cdot \mathbf{v}| = 3\sqrt{2}$$
9. $$|\mathbf{v}| = \sqrt{2^2 + 2^2} = 2\sqrt{2}$$
10. $$|4 \cdot \mathbf{v}| = 4 \cdot 5 = 20$$
11. $$|\mathbf{v}| = 25$$
12. $$|-3 \cdot \mathbf{v}| = 3\sqrt{10}$$
13. $$|\mathbf{v}| = \sqrt{8^2 + 15^2} = 17$$
14. $$|\mathbf{v}| = \sqrt{4^2 + 3^2} = 5$$
15. $$|\mathbf{v}| = \sqrt{(-2)^2 + (-6)^2} = \sqrt{40} = 2\sqrt{10}$$
16. $$|\mathbf{v}| = 10$$
17. $$|3 \cdot \mathbf{v}| = 3\sqrt{5}$$
18. $$|\mathbf{v}| = 9$$
19. $$|5 \cdot \mathbf{v}| = 10\sqrt{2}$$
20. $$|\mathbf{v}| = 15$$

Medium Level Answers

1. $$|\mathbf{v}| = \sqrt{2^2 + 3^2} = \sqrt{13}$$; $$|2 \cdot \mathbf{v}| = 2\sqrt{13}$$
2. $$|\mathbf{v}| = \sqrt{(-4)^2 + 3^2} = 5$$
3. $$|k \cdot \mathbf{v}| = 3$$
4. $$|\mathbf{v}| = 10$$; $$|3 \cdot \mathbf{v}| = 30$$
5. $$|\mathbf{v}| = 13$$
6. $$|\mathbf{v}| = \sqrt{9^2 + (-40)^2} = 41$$
7. $$|2 \cdot \mathbf{v}| = 2\sqrt{2}$$
8. $$|\mathbf{v}| = 5$$
9. $$|\mathbf{v}| = \sqrt{7^2 + 1^2} = \sqrt{50} = 5\sqrt{2}$$
10. $$|\mathbf{v}| = 8$$
11. $$|5 \cdot \mathbf{v}| = 10$$
12. $$|3 \cdot \mathbf{v}| = 12$$
13. $$|\mathbf{v}| = \sqrt{(-7)^2 + (-24)^2} = 25$$
14. $$|\mathbf{v}| = \sqrt{2^2 + (-2)^2} = 2\sqrt{2}$$
15. $$|\mathbf{v}| = 14.14$$
16. $$|k \cdot \mathbf{v}| = 6$$
17. $$|\mathbf{v}| = \sqrt{0^2 + 8^2} = 8$$
18. $$|4 \cdot \mathbf{v}| = 40$$
19. $$|\mathbf{v}| = 1$$
20. $$|4 \cdot \mathbf{v}| = 24$$

Hard Level Answers

1. $$|k \cdot \mathbf{v}| = 10$$
2. $$|\mathbf{v}| = 17$$; $$|k \cdot \mathbf{v}| = 51$$
3. $$|\mathbf{v}| = 13$$; $$|k \cdot \mathbf{v}| = 39$$
4. $$|\mathbf{v}| = \sqrt{2}$$; $$|k \cdot \mathbf{v}| = 6$$
5. $$|\mathbf{v}| = 25$$; $$|k \cdot \mathbf{v}| = 25$$
6. $$|\mathbf{v}| = 10$$; $$|k \cdot \mathbf{v}| = 40$$
7. $$|\mathbf{v}| = 9$$; $$|k \cdot \mathbf{v}| = 45$$
8. $$|\mathbf{v}| = 2\sqrt{2}$$; $$|k \cdot \mathbf{v}| = 0$$
9. $$|\mathbf{v}| = 50$$; $$|k \cdot \mathbf{v}| = 100$$
10. $$|\mathbf{v}| = 5$$; $$|k \cdot \mathbf{v}| = 15$$
11. $$|\mathbf{v}| = 13$$; $$|k \cdot \mathbf{v}| = 52$$
12. $$|\mathbf{v}| = 25$$; $$|k \cdot \mathbf{v}| = 75$$
13. $$|\mathbf{v}| = 8$$; $$|k \cdot \mathbf{v}| = 16$$
14. $$|\mathbf{v}| = 14.14$$; $$|k \cdot \mathbf{v}| = 28.28$$
15. $$|\mathbf{v}| = 10$$; $$|k \cdot \mathbf{v}| = 40$$
16. $$|\mathbf{v}| = 2\sqrt{2}$$; $$|k \cdot \mathbf{v}| = 8\sqrt{2}