Problem 8 - Calculus: Early Transcendentals

Question

Find the cosine of the angle between \langle 47,100,0\rangle and \langle 0,0,5\rangle\(;\) use a calculator if necessary to find the angle.

Step-by-Step Solution

Answer

The vectors are perpendicular; the angle between them is 90 degrees.

Step 1: Understand the Dot Product Formula

The dot product of two vectors \( \langle a_1, a_2, a_3 \rangle \) and \( \langle b_1, b_2, b_3 \rangle \) is given by \( a_1b_1 + a_2b_2 + a_3b_3 \). We want to find the angle between these vectors using the formula for cosine, which is \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|} \).

Step 2: Calculate the Dot Product

For the vectors \( \langle 47, 100, 0 \rangle \) and \( \langle 0, 0, 5 \rangle \), the dot product is calculated as follows: \( 47 \times 0 + 100 \times 0 + 0 \times 5 = 0 \).

Step 3: Calculate Magnitudes of the Vectors

The magnitude of a vector \( \langle a_1, a_2, a_3 \rangle \) is given by \( \sqrt{a_1^2 + a_2^2 + a_3^2} \). Thus, the magnitude of \( \langle 47, 100, 0 \rangle \) is \( \sqrt{47^2 + 100^2 + 0^2} = \sqrt{2209 + 10000} = \sqrt{12209} \). The magnitude of \( \langle 0, 0, 5 \rangle \) is \( \sqrt{0^2 + 0^2 + 5^2} = \sqrt{25} = 5 \).

Step 4: Substitute into Cosine Formula

Substitute the dot product and magnitudes into the cosine formula: \( \cos \theta = \frac{0}{\sqrt{12209} \times 5} = 0 \).

Step 5: Determine the Angle

Since \( \cos \theta = 0 \), the angle \( \theta \) is \( 90^\circ \) or \( \frac{\pi}{2} \) radians. This indicates that the vectors are perpendicular.

Key Concepts

Dot Product Formula

The dot product, also known as the scalar product, is a way to multiply two vectors and get a scalar (a single number) rather than another vector. To calculate the dot product of two vectors, say \( \langle a_1, a_2, a_3 \rangle \) and \( \langle b_1, b_2, b_3 \rangle \), you multiply their corresponding components and add them up. Here's the formula:

• \( a_1b_1 + a_2b_2 + a_3b_3 \)

In this exercise, our vectors are \( \langle 47, 100, 0 \rangle \) and \( \langle 0, 0, 5 \rangle \). When computing the dot product of these vectors, you get:

• \( 47 \times 0 + 100 \times 0 + 0 \times 5 = 0 \)

Notice the result is zero. This specific result can tell us that the vectors are perpendicular or orthogonal to each other.

Magnitude of a Vector

The magnitude (or length) of a vector measures how long the vector is. You can think of it as the distance from the origin to the point represented by the vector in a coordinate system. To find the magnitude of a vector \( \langle a_1, a_2, a_3 \rangle \), use the following formula:

• \( \sqrt{a_1^2 + a_2^2 + a_3^2} \)

Let's calculate the magnitudes for the vectors in this problem:For \( \langle 47, 100, 0 \rangle \), the magnitude is:

• \( \sqrt{47^2 + 100^2 + 0^2} = \sqrt{12209} \)

This number doesn't need to be exact for this exercise, but it tells us how 'long' the vector is. For \( \langle 0, 0, 5 \rangle \), the magnitude is:

• \( \sqrt{0^2 + 0^2 + 5^2} = 5 \)

Again, it's a measure of how long or big this vector is.

Cosine Formula

The cosine formula helps us find the angle between two vectors. This is done by dividing the dot product of the vectors by the product of their magnitudes.Here's the formula:

• \( \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|\|\mathbf{b}\|} \)

For our vectors \( \langle 47, 100, 0 \rangle \) and \( \langle 0, 0, 5 \rangle \), we already calculated:

• Dot product = 0
• Magnitude of \( \langle 47, 100, 0 \rangle = \sqrt{12209} \)
• Magnitude of \( \langle 0, 0, 5 \rangle = 5 \)

Substituting into the cosine formula:

• \( \cos \theta = \frac{0}{\sqrt{12209} \times 5} = 0 \)

Since \( \cos \theta = 0 \), the angle \( \theta \) is \( 90^\circ \) or \( \frac{\pi}{2} \) radians, showing the vectors are perpendicular.