Problem 16 - APEX Calculus

Question

Find the measure of the angle between the two vectors in both radians and degrees. \(\vec{u}=\langle 1,7,2\rangle, \vec{v}=\langle 4,-2,5\rangle\)

Step-by-Step Solution

Answer

The angle between the vectors is \( \frac{\pi}{2} \) radians or 90 degrees.

Step 1: Calculate the Dot Product

To find the angle between two vectors \( \vec{u} \) and \( \vec{v} \), we first compute their dot product. The formula for the dot product is:\[ \vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3 \]Substituting the vector components, we have:\[ \vec{u} \cdot \vec{v} = 1 \cdot 4 + 7 \cdot (-2) + 2 \cdot 5 = 4 - 14 + 10 = 0 \]

Step 2: Calculate the Magnitude of Each Vector

Now, calculate the magnitude of each vector. The magnitude of a vector \( \vec{u} = \langle u_1, u_2, u_3 \rangle \) is given by:\[ \| \vec{u} \| = \sqrt{u_1^2 + u_2^2 + u_3^2} \]For \( \vec{u} = \langle 1, 7, 2 \rangle \):\[ \| \vec{u} \| = \sqrt{1^2 + 7^2 + 2^2} = \sqrt{1 + 49 + 4} = \sqrt{54} \]For \( \vec{v} = \langle 4, -2, 5 \rangle \):\[ \| \vec{v} \| = \sqrt{4^2 + (-2)^2 + 5^2} = \sqrt{16 + 4 + 25} = \sqrt{45} \]

Step 3: Use the Dot Product to Find the Cosine of the Angle

The cosine of the angle \( \theta \) between two vectors is given by the formula:\[ \cos \theta = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \|\vec{v}\|} \]Substituting the values from Steps 1 and 2:\[ \cos \theta = \frac{0}{\sqrt{54} \cdot \sqrt{45}} = 0 \]

Step 4: Determine the Angle in Radians and Degrees

Since \( \cos \theta = 0 \), the angle \( \theta \) is \( \frac{\pi}{2} \) radians because cosine of \( \pi/2 \) is zero.To convert radians to degrees, use the conversion:\[ \theta = \frac{\pi}{2} \times \frac{180}{\pi} = 90 \text{ degrees} \]

Key Concepts

Dot Product

The dot product is a fundamental concept in vector mathematics. It is an operation that takes two vectors and returns a single number, often referred to as a scalar. The formula for the dot product of two vectors \(\vec{u} = \langle u_1, u_2, u_3 \rangle\) and \(\vec{v} = \langle v_1, v_2, v_3 \rangle\) is given by:

• \(\vec{u} \cdot \vec{v} = u_1v_1 + u_2v_2 + u_3v_3\)

This formula involves multiplying corresponding components of the vectors and summing the results. In this exercise, the dot product of vectors \(\vec{u} = \langle 1, 7, 2 \rangle\) and \(\vec{v} = \langle 4, -2, 5 \rangle\) results in zero, indicating that the vectors are perpendicular.

Magnitude of a Vector

The magnitude of a vector measures its length. It is often referred to as the vector's "norm" or "length." For a vector \(\vec{u} = \langle u_1, u_2, u_3 \rangle\), the magnitude is calculated using the formula:

• \(\| \vec{u} \| = \sqrt{u_1^2 + u_2^2 + u_3^2}\)

This formula finds the square root of the sum of the squares of the vector's components. In our example, the magnitudes of vectors \(\vec{u}\) and \(\vec{v}\) are \(\sqrt{54}\) and \(\sqrt{45}\), respectively.
Calculating magnitude is crucial when determining the angle between vectors using their dot product.

Cosine of the Angle

The cosine of the angle between two vectors is crucial to find the precise angle between them. This is computed using the dot product and the magnitudes of the vectors. The relevant formula is:

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

In this exercise, we found that the dot product is zero, which gives us \(\cos \theta = 0\). This result indicates that the vectors are perpendicular, as the cosine of \(90^\circ\) (or \(\frac{\pi}{2}\) radians) is zero.
Understanding how to calculate the cosine of the angle will allow you to compare the directions of different vectors.

Radian and Degree Conversion

Angles can be measured in degrees or radians. It's often necessary in mathematics to convert between these units. One full circle is \(360\) degrees, which is equivalent to \(2\pi\) radians. To convert radians to degrees, use the formula:

• \(\theta~\text{(in degrees)} = \theta~\text{(in radians)} \times \frac{180}{\pi}\)

In our problem, since the angle \(\theta\) calculated was \(\frac{\pi}{2}\) radians, multiplying \(\frac{\pi}{2}\) by \(\frac{180}{\pi}\) yields \(90\) degrees. Knowing how to convert between these units is vital when working with different applications in mathematics and engineering.
Keep this conversion handy to switch between angle measurements effortlessly.