Trigonometry Vectors Worksheet – Step-by-Step Guidance

Q1. Sketch a graph of the vector \( \vec{r} = \langle -4, -4 \rangle \)

Background

Topic: Vectors in the Coordinate Plane

This question is testing your ability to represent a vector in component form by graphing it on the coordinate plane.

Key Terms:

• Component form: A vector written as \( \langle a, b \rangle \), where \( a \) is the horizontal (x) component and \( b \) is the vertical (y) component.

• Graphing a vector: Draw an arrow from the origin (0,0) to the point (a, b).

Step-by-Step Guidance

1. Start at the origin (0, 0) on the coordinate plane.

2. Move left by 4 units (since the x-component is -4).

3. Move down by 4 units (since the y-component is -4).

4. Draw an arrow from (0, 0) to the point (-4, -4). This represents the vector \( \vec{r} \).

Try sketching the vector before checking your work!

Q2. Sketch a graph of the vector \( \vec{b} = \langle 19, -2 \rangle \)

Background

Topic: Vectors in the Coordinate Plane

This question is about plotting a vector given its components.

Key Terms:

• Component form: \( \langle a, b \rangle \)

Step-by-Step Guidance

1. Start at the origin (0, 0).

2. Move right by 19 units (x-component is positive).

3. Move down by 2 units (y-component is negative).

4. Draw an arrow from (0, 0) to (19, -2).

Try sketching the vector before checking your work!

Q3. Given \( \vec{u} = \langle -2, 2 \rangle \) and \( \vec{v} = \langle -5, -4 \rangle \), find \( \vec{u} + \vec{v} \) in component form.

Background

Topic: Vector Addition

This question tests your ability to add vectors in component form.

Key Formula:

To add two vectors:

$\vec{u} + \vec{v} = \langle u_1 + v_1,\ u_2 + v_2 \rangle$

Step-by-Step Guidance

1. Write the components of each vector: \( \vec{u} = \langle -2, 2 \rangle \), \( \vec{v} = \langle -5, -4 \rangle \).

2. Add the x-components: $-2 + (-5)$.

3. Add the y-components: $2 + (-4)$.

4. Combine your results into a new vector: $\langle \text{sum of x-components},\ \text{sum of y-components} \rangle$.

Try calculating the sums before checking your answer!

Q4. Given \( \vec{f} = \langle -6, 3 \rangle \) and \( \vec{g} = \langle -1, -3 \rangle \), find \( \vec{f} + \vec{g} \) in component form.

Background

Topic: Vector Addition

This question is about adding two vectors by adding their corresponding components.

Key Formula:

$\vec{f} + \vec{g} = \langle f_1 + g_1,\ f_2 + g_2 \rangle$

Step-by-Step Guidance

1. Write the components: \( \vec{f} = \langle -6, 3 \rangle \), \( \vec{g} = \langle -1, -3 \rangle \).

2. Add the x-components: $-6 + (-1)$.

3. Add the y-components: $3 + (-3)$.

4. Express the result as a vector: $\langle \text{sum of x-components},\ \text{sum of y-components} \rangle$.

Try adding the components before checking your answer!

Q5. Given \( \vec{u} = \langle -9, 3 \rangle \), find \( -4\vec{u} \).

Background

Topic: Scalar Multiplication of Vectors

This question tests your ability to multiply a vector by a scalar.

Key Formula:

$k\vec{u} = \langle k u_1,\ k u_2 \rangle$

Step-by-Step Guidance

1. Identify the scalar: $k = -4$.

2. Multiply the x-component: $-4 \times (-9)$.

3. Multiply the y-component: $-4 \times 3$.

4. Write the new vector as $\langle \text{result for x},\ \text{result for y} \rangle$.

Try multiplying before checking your answer!

Q6. Given \( \vec{f} = \langle -12, 6 \rangle \), find \( 8\vec{f} \).

Background

Topic: Scalar Multiplication of Vectors

This question is about multiplying each component of a vector by a scalar.

Key Formula:

$k\vec{f} = \langle k f_1,\ k f_2 \rangle$

Step-by-Step Guidance

1. Identify the scalar: $k = 8$.

2. Multiply the x-component: $8 \times (-12)$.

3. Multiply the y-component: $8 \times 6$.

4. Write the result as $\langle \text{result for x},\ \text{result for y} \rangle$.

Try multiplying before checking your answer!

Q7. Find the magnitude of the vector \( \vec{p} = \langle -4, 8 \rangle \).

Background

Topic: Magnitude of a Vector

This question is about finding the length of a vector using the Pythagorean Theorem.

Key Formula:

$|\vec{p}| = \sqrt{p_1^2 + p_2^2}$

Step-by-Step Guidance

1. Identify the components: $p_1 = -4$, $p_2 = 8$.

2. Square each component: $(-4)^2$ and $8^2$.

3. Add the squares together: $(-4)^2 + 8^2$.

4. Take the square root of the sum to find the magnitude.

Try calculating the magnitude before checking your answer!

Q8. Find the magnitude of the vector \( \vec{v} = \langle -21, 28 \rangle \).

Background

Topic: Magnitude of a Vector

This question is about using the distance formula to find the length of a vector.

Key Formula:

$|\vec{v}| = \sqrt{(-21)^2 + 28^2}$

Step-by-Step Guidance

1. Square each component: $(-21)^2$ and $28^2$.

2. Add the results: $(-21)^2 + 28^2$.

3. Take the square root of the sum to find the magnitude.

Try calculating the magnitude before checking your answer!

Q9. Make a sketch, then find the direction angle for \( \vec{b} = \langle -9, -40 \rangle \).

Background

Topic: Direction Angle of a Vector

This question is about finding the angle a vector makes with the positive x-axis, measured counterclockwise.

Key Formula:

$\theta = \arctan\left(\frac{y}{x}\right)$

Be careful with the signs and the quadrant when interpreting the angle.

Step-by-Step Guidance

1. Identify the components: $x = -9$, $y = -40$.

2. Calculate $\arctan\left(\frac{y}{x}\right) = \arctan\left(\frac{-40}{-9}\right)$.

3. Determine the reference angle using your calculator (in degrees).

4. Since both components are negative, the vector is in the third quadrant. Adjust the angle accordingly to find the direction angle measured from the positive x-axis.

Try finding the angle before checking your answer!

Q10. Make a sketch, then find the direction angle for \( \vec{m} = \langle -1, 10 \rangle \).

Background

Topic: Direction Angle of a Vector

This question is about finding the angle a vector makes with the positive x-axis.

Key Formula:

$\theta = \arctan\left(\frac{y}{x}\right)$

Step-by-Step Guidance

1. Identify the components: $x = -1$, $y = 10$.

2. Calculate $\arctan\left(\frac{10}{-1}\right)$.

3. Find the reference angle using your calculator (in degrees).

4. Since x is negative and y is positive, the vector is in the second quadrant. Adjust the angle to reflect this.

Try finding the angle before checking your answer!

Q11. A twin-engine airplane has a speed of 300 mi/h in still air. Suppose this airplane heads directly south and encounters a 50 mi/h wind blowing due east. Find the resulting speed and direction of the plane. Round your answers to the nearest unit.

Background

Topic: Vector Addition (Application)

This question is about combining two perpendicular vectors (airplane velocity and wind velocity) to find the resultant vector's magnitude and direction.

Key Formulas:

• Resultant vector: $\vec{v}_r = \vec{v}_1 + \vec{v}_2$

• Magnitude: $|\vec{v}_r| = \sqrt{v_{rx}^2 + v_{ry}^2}$

• Direction angle: $\theta = \arctan\left(\frac{v_{rx}}{v_{ry}}\right)$ (relative to south, or adjust as needed)

Step-by-Step Guidance

1. Express the airplane's velocity as a vector: since it heads south, its components are $(0, -300)$ (assuming north is positive y).

2. Express the wind's velocity as a vector: $(50, 0)$ (since it blows due east).

3. Add the vectors: $(0 + 50, -300 + 0) = (50, -300)$.

4. Find the magnitude: $\sqrt{50^2 + (-300)^2}$.

5. Find the direction angle: $\theta = \arctan\left(\frac{50}{-300}\right)$ (relative to due south or the positive x-axis, as appropriate).

Try calculating the magnitude and angle before checking your answer!