Problem 83 - Precalculus

Question

A rectangle has sides with lengths 18 units and 11 units. Find the angle to one decimal place between the diagonal and the side with length of 18 units. (Hint: Set up a rectangular coordinate system, and use vectors (18,0) to represent the side of length 18 units and \langle 18,11\rangle to represent the diagonal.)

Step-by-Step Solution

Answer

The angle between the diagonal and the 18-unit side is approximately 31.8 degrees.

Step 1: Understand Vectors

We are given a rectangle where one side has a vector representation as \( \vec{a} = \langle 18, 0 \rangle \) along the x-axis. The diagonal is identified through vector \( \vec{d} = \langle 18, 11 \rangle \). Our task is to determine the angle between these two vectors.

Step 2: Apply the Dot Product Formula

The dot product of two vectors \( \vec{a} = \langle a_1, a_2 \rangle \) and \( \vec{b} = \langle b_1, b_2 \rangle \) is given by \( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 \). Here, we compute the dot product of \( \vec{a} \) and \( \vec{d} \): \( \vec{a} \cdot \vec{d} = 18 \times 18 + 0 \times 11 = 324 \).

Step 3: Calculate Magnitudes of Vectors

The magnitude of a vector \( \vec{a} = \langle a_1, a_2 \rangle \) is given by \( |\vec{a}| = \sqrt{a_1^2 + a_2^2} \). For \( \vec{a} \), \( |\vec{a}| = \sqrt{18^2 + 0^2} = 18 \). For diagonal \( \vec{d} \), \( |\vec{d}| = \sqrt{18^2 + 11^2} = \sqrt{324 + 121} = \sqrt{445} \).

Step 4: Use the Angle Formula Between Two Vectors

The cosine of the angle \( \theta \) between two vectors \( \vec{a} \) and \( \vec{b} \) is given by \( \cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \times |\vec{b}|} \). Substitute the values: \( \cos \theta = \frac{324}{18 \times \sqrt{445}} \). Simplify the denominator: \( 18 \times \sqrt{445} \approx 379.08 \). Thus, \( \cos \theta \approx \frac{324}{379.08} \approx 0.8544 \).

Step 5: Calculate the Angle in Degrees

Use the inverse cosine function to determine \( \theta \): \( \theta = \cos^{-1}(0.8544) \). Computing this gives \( \theta \approx 31.8 \) degrees. This is the angle between the diagonal and the 18-unit side.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. It allows us to establish a relationship between two vectors.Here’s how it works:

• Given two vectors, \( \vec{a} = \langle a_1, a_2 \rangle \) and \( \vec{b} = \langle b_1, b_2 \rangle \), their dot product is calculated as \( \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 \).
• This operation results in a scalar, which reveals something about how much one vector extends in the direction of the other.

In the context of the rectangle, the side vector is \( \vec{a} = \langle 18, 0 \rangle \) and the diagonal vector is \( \vec{d} = \langle 18, 11 \rangle \).By plugging in these values, we got the dot product:\[ \vec{a} \cdot \vec{d} = 18 \times 18 + 0 \times 11 = 324 \] The result, 324, serves as a component in calculating the angle between the vectors later.

Magnitude of Vectors

Magnitude is like the length or size of the vector. Picture it as the distance from the origin to a point defined by the vector.To calculate it:

• For a vector \( \vec{a} = \langle a_1, a_2 \rangle \), its magnitude \( |\vec{a}| \) is given by \( \sqrt{a_1^2 + a_2^2} \).

When applied to our rectangle example, the side vector \( \vec{a} = \langle 18, 0 \rangle \) has a magnitude of:\[ |\vec{a}| = \sqrt{18^2 + 0^2} = 18 \]Next, the magnitude of the diagonal vector \( \vec{d} = \langle 18, 11 \rangle \) is:\[ |\vec{d}| = \sqrt{18^2 + 11^2} = \sqrt{445} \]The magnitude helps us appreciate the actual "length" of the vector, crucial for finding angles.

Angle Between Vectors

Understanding the angle between vectors is important in geometry and physics, as it determines how one vector is oriented relative to another.Here's how you can find this angle:

• Using the formula \( \cos \theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \times |\vec{b}|} \), you can calculate the cosine of the angle between two vectors.
• The angle \( \theta \) can then be found by taking the inverse cosine.

For our example, we've already calculated:

• \( \vec{a} \cdot \vec{d} = 324 \)
• \( |\vec{a}| = 18 \)
• \( |\vec{d}| = \sqrt{445} \)

Now, substitute these into the angle formula:\[ \cos \theta = \frac{324}{18 \times \sqrt{445}} \approx 0.8544 \]Finally, the angle \( \theta \) is:\[ \theta = \cos^{-1}(0.8544) \approx 31.8 \text{ degrees} \]This confirms the orientation of the diagonal in relation to the 18-unit side, bringing with it a deeper understanding of vector operations in geometry.