Problem 7 - Algebra and Trigonometry

Question

Find (a) \(\mathbf{u} \cdot \mathbf{v}\) and (b) the angle between \(\mathbf{u}\) and \(\mathbf{v}\) to the nearest degree. $$ \mathbf{u}=\langle 2,7\rangle, \quad \mathbf{v}=\langle 3,1\rangle $$

Step-by-Step Solution

Answer

(a) \( \mathbf{u} \cdot \mathbf{v} = 13 \), (b) Angle \( \approx 57^\circ \).

Step 1: Compute Dot Product

To find \( \mathbf{u} \cdot \mathbf{v} \), use the formula for the dot product of two vectors: \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \). Substitute the components of \( \mathbf{u} = \langle 2, 7 \rangle \) and \( \mathbf{v} = \langle 3, 1 \rangle \):\[ \mathbf{u} \cdot \mathbf{v} = (2)(3) + (7)(1) = 6 + 7 = 13 \]

Step 2: Find Magnitude of Vectors

The magnitude of vector \( \mathbf{u} = \langle 2, 7 \rangle \) is given by \( \| \mathbf{u} \| = \sqrt{u_1^2 + u_2^2} = \sqrt{2^2 + 7^2} = \sqrt{4 + 49} = \sqrt{53} \).The magnitude of vector \( \mathbf{v} = \langle 3, 1 \rangle \) is given by \( \| \mathbf{v} \| = \sqrt{v_1^2 + v_2^2} = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \).

Step 3: Use Dot Product Formula to Find Angle

The formula to find the angle \( \theta \) between two vectors using their dot product is:\[ \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\| \mathbf{u} \| \| \mathbf{v} \|} \]Substitute the values: \( \cos \theta = \frac{13}{\sqrt{53} \times \sqrt{10}} \).Simplify the denominator: \( \cos \theta = \frac{13}{\sqrt{530}} \).

Step 4: Compute the Angle

Calculate \( \theta = \cos^{-1}\left(\frac{13}{\sqrt{530}}\right) \) using a calculator to find the angle in degrees.It comes out to approximately \( \theta = 57^\circ \).

Key Concepts

Understanding the Dot Product

The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It combines two vectors to yield a single scalar value. This operation is quite simple when we break it down.
To compute the dot product of vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), use this formula:

• \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \)

When you multiply the corresponding components of the vectors and then add them up, you get the dot product.
For example, given \( \mathbf{u} = \langle 2, 7 \rangle \) and \( \mathbf{v} = \langle 3, 1 \rangle \), substitute these into the formula to find: \[ \mathbf{u} \cdot \mathbf{v} = (2)(3) + (7)(1) = 6 + 7 = 13 \] In this exercise, the dot product equals 13. This scalar represents how much one vector goes in the direction of another. The dot product is particularly useful in physics for calculating work done or in computer graphics for lighting calculations.

Calculating the Magnitude of a Vector

The magnitude of a vector, also referred to as the vector's length or norm, is a measure of how long the vector is. To find the magnitude of a vector, use the following formula:

• \( \| \mathbf{u} \| = \sqrt{u_1^2 + u_2^2} \)

To see this in practice, let's calculate the magnitude for two vectors \( \mathbf{u} = \langle 2, 7 \rangle \) and \( \mathbf{v} = \langle 3, 1 \rangle \):

• For \( \mathbf{u} \): \( \| \mathbf{u} \| = \sqrt{2^2 + 7^2} = \sqrt{4 + 49} = \sqrt{53} \)
• For \( \mathbf{v} \): \( \| \mathbf{v} \| = \sqrt{3^2 + 1^2} = \sqrt{9 + 1} = \sqrt{10} \)

The magnitude tells you the length or size of the vector and plays an essential role when determining the distance between two points or calculating physical quantities like velocity and force in real-world applications.

Finding the Angle Between Vectors

Finding the angle between two vectors is crucial in many fields such as physics, computer science, and engineering. To determine this angle, you need the dot product and the magnitudes of the vectors.
The formula used to find the cosine of the angle \( \theta \) between vectors \( \mathbf{u} \) and \( \mathbf{v} \) is:

• \( \cos \theta = \frac{\mathbf{u} \cdot \mathbf{v}}{\| \mathbf{u} \| \| \mathbf{v} \|} \)

For the given vectors \( \mathbf{u} = \langle 2, 7 \rangle \) and \( \mathbf{v} = \langle 3, 1 \rangle \), substitute the known values:
\[ \cos \theta = \frac{13}{\sqrt{530}} \] Once you have \( \cos \theta \), use a calculator to find the angle \( \theta \) itself. Converting the result from cos inverse gives you the angle in degrees, which is approximately 57°. Knowing this angle can help describe how vectors interact with each other in space, assisting in the orientation and alignment tasks.