Problem 15 - APEX Calculus

Question

Find the measure of the angle between the two vectors in both radians and degrees. \(\vec{u}=\langle 8,1,-4\rangle, \vec{v}=\langle 2,2,0\rangle\)

Step-by-Step Solution

Answer

The angle between the vectors is \(\frac{\pi}{4}\) radians or 45 degrees.

Step 1: Calculate the Dot Product of the Vectors

The dot product of two vectors \(\vec{u} = \langle a_1, b_1, c_1 \rangle\) and \(\vec{v} = \langle a_2, b_2, c_2 \rangle\) is given by \(\vec{u} \cdot \vec{v} = a_1 \cdot a_2 + b_1 \cdot b_2 + c_1 \cdot c_2\). For the vectors \(\vec{u} = \langle 8, 1, -4 \rangle\) and \(\vec{v} = \langle 2, 2, 0 \rangle\), this becomes \(\vec{u} \cdot \vec{v} = 8 \cdot 2 + 1 \cdot 2 + (-4) \cdot 0 = 16 + 2 + 0 = 18\).

Step 2: Calculate the Magnitude of Each Vector

The magnitude of a vector \(\vec{u} = \langle a, b, c \rangle\) is found using the formula \(\|\vec{u}\| = \sqrt{a^2 + b^2 + c^2}\). For \(\vec{u} = \langle 8, 1, -4 \rangle\), we have \(\|\vec{u}\| = \sqrt{8^2 + 1^2 + (-4)^2} = \sqrt{64 + 1 + 16} = \sqrt{81} = 9\). For \(\vec{v} = \langle 2, 2, 0 \rangle\), \(\|\vec{v}\| = \sqrt{2^2 + 2^2 + 0^2} = \sqrt{4 + 4 + 0} = \sqrt{8}\) which simplifies to \(2\sqrt{2}\).

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 \(\cos(\theta) = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\| \| \vec{v} \|}\). Substituting in the values from Steps 1 and 2, we get \(\cos(\theta) = \frac{18}{9 \times 2\sqrt{2}} = \frac{18}{18\sqrt{2}} = \frac{1}{\sqrt{2}}\).

Step 4: Find the Angle in Radians

The angle in radians, \(\theta\), can be found by taking the inverse cosine of the value obtained for \(\cos(\theta)\). Hence, \(\theta = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4}\) radians, as \(\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\).

Step 5: Convert the Angle to Degrees

To convert radians to degrees, use the conversion factor \(180/\pi\). Therefore, for an angle of \(\frac{\pi}{4}\) radians, the angle in degrees is \(\frac{\pi}{4} \times \frac{180}{\pi} = 45\) degrees.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector geometry, helping to find the relationship between two vectors. It is an algebraic operation that combines two vectors and returns a single number, which can indicate whether the vectors are orthogonal, parallel, or neither.

To find the dot product of two vectors, we use the formula:

• For vectors \( \vec{u} = \langle a_1, b_1, c_1 \rangle \) and \( \vec{v} = \langle a_2, b_2, c_2 \rangle \), the dot product is \( \vec{u} \cdot \vec{v} = a_1 \cdot a_2 + b_1 \cdot b_2 + c_1 \cdot c_2 \).

For example, the dot product of the vectors \(\vec{u} = \langle 8, 1, -4 \rangle\) and \(\vec{v} = \langle 2, 2, 0 \rangle\)is \(8 \cdot 2 + 1 \cdot 2 + (-4) \cdot 0 = 18\).

This product is significant; a positive dot product indicates that the angle between the vectors is less than 90 degrees, while a negative product would suggest an angle greater than 90 degrees.

Magnitude of Vectors

Magnitude gives us the length of a vector and is an essential aspect of understanding vector size and orientation. It's akin to finding the hypotenuse in a 3D space from the vector components.

The magnitude of a vector \(\vec{u} = \langle a, b, c \rangle\) is calculated using:

• \(\|\vec{u}\| = \sqrt{a^2 + b^2 + c^2}\)

Let's look at the vectors in our example:

• The magnitude of \( \vec{u} = \langle 8, 1, -4 \rangle \) is \( \|\vec{u}\| = \sqrt{8^2 + 1^2 + (-4)^2} = 9 \).
• For \( \vec{v} = \langle 2, 2, 0 \rangle \), the magnitude is \( \|\vec{v}\| = \sqrt{2^2 + 2^2 + 0^2} = 2\sqrt{2} \).

These magnitudes are crucial when finding the angle between vectors since they normalize the scale of each vector.

Angle Between Vectors

Understanding the angle between two vectors is crucial in physics and engineering, where the direction may affect forces, velocity, or similar aspects. The angle can be found using the cosine of the angle \(\theta\)between vectors.

The formula is:

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

Substituting the values from our example, we get:

• \( \cos(\theta) = \frac{18}{9 \times 2\sqrt{2}} \)
• This simplifies to \( \frac{1}{\sqrt{2}} \).

Finally, to find the angle \(\theta\), we use the inverse cosine:

• \( \theta = \cos^{-1}\left(\frac{1}{\sqrt{2}}\right) = \frac{\pi}{4} \) radians.

This calculation reveals that the vectors form a 45-degree angle with each other.

Radian to Degree Conversion

Angles can be expressed in radians or degrees, offering flexibility in computations based on context. Radian is a standard unit in mathematics, but degrees are more intuitive in day-to-day applications.

To convert radians to degrees, employ the conversion factor: \(\frac{180}{\pi}\). This is because a full circle is equal to \(2\pi\) radians or 360 degrees.

In the exercise at hand:

• The angle \( \frac{\pi}{4} \) radians converts to degrees as follows:
• Multiply by \( \frac{180}{\pi} \)
• Resulting in: \( \frac{\pi}{4} \times \frac{180}{\pi} = 45 \) degrees.

Knowing how to convert between these units is vital for interpreting and communicating angle measurements effectively across various scientific and engineering disciplines.