Problem 86 - Precalculus

Question

Use the graphing calculator \([\mathrm{sum}]\) command to find the angle (round to the nearest degree) between each pair of vectors. $$(-12,9),(-21,-13)$$

Step-by-Step Solution

Answer

The angle between the vectors is approximately \(81^\circ\).

Step 1: Find the Dot Product

The dot product of two vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \) is given by:\[a \cdot b = a_1b_1 + a_2b_2\]For our vectors \((-12,9)\) and \((-21,-13)\):\[a \cdot b = (-12)(-21) + (9)(-13) = 252 - 117 = 135\]

Step 2: Find the Magnitudes of Each Vector

Calculate the magnitude of each vector using the formula:\[||\mathbf{a}|| = \sqrt{a_1^2 + a_2^2}\]For \((-12, 9)\):\[||\mathbf{a}|| = \sqrt{(-12)^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15\]For \((-21, -13)\):\[||\mathbf{b}|| = \sqrt{(-21)^2 + (-13)^2} = \sqrt{441 + 169} = \sqrt{610}\]

Step 3: Use Cosine Formula for Angle Between Vectors

The cosine of the angle \( \theta \) between two vectors is given by:\[\cos \theta = \frac{a \cdot b}{||\mathbf{a}|| \cdot ||\mathbf{b}||}\]Substituting the values from Step 1 and 2:\[\cos \theta = \frac{135}{15 \times \sqrt{610}} = \frac{135}{15\sqrt{610}} = \frac{9}{\sqrt{610}}\]

Step 4: Calculate the Angle

To find the angle \( \theta \), we take the inverse cosine:\[\theta = \cos^{-1}\left(\frac{9}{\sqrt{610}}\right)\]Using a calculator, \( \theta \approx 81.1^\circ \). Rounding to the nearest degree, the angle is \(81^\circ\).

Key Concepts

The Magnitude of a Vector

The magnitude of a vector is a measure of its length. To find the magnitude of a two-dimensional vector \((a_1, a_2)\), you can use the formula:

• \[ ||\mathbf{a}|| = \sqrt{a_1^2 + a_2^2} \]

This formula itself comes from the Pythagorean theorem. It's like finding the hypotenuse of a right triangle where the vector's components form the other sides.
For example, with vector \((-12, 9)\):

• Calculate its magnitude: \[ ||\mathbf{a}|| = \sqrt{(-12)^2 + 9^2} = 15 \]

This tells us the vector has a length of 15 units.The magnitude helps in understanding the size or extent of the vector from the origin to its point in the coordinate system. Always keep in mind, magnitude is always a non-negative value.

Cosine of the Angle Between Vectors

To find the angle between two vectors, we make use of the cosine formula. The cosine of the angle \( \theta \) between vectors \( \mathbf{a} \) and \( \mathbf{b} \) can be calculated by:

• \[ \cos \theta = \frac{a \cdot b}{||\mathbf{a}|| \cdot ||\mathbf{b}||} \]

Here, \( a \cdot b \) represents the dot product, while \( ||\mathbf{a}|| \) and \( ||\mathbf{b}|| \) are the magnitudes of the vectors.
The dot product measures how much two vectors "point" in the same direction.

• For the vectors \((-12,9)\) and \((-21,-13)\), we calculated the dot product to be 135.

Finding the cosine tells us about the closeness in direction between the vectors. An angle close to 0° means the vectors are pointing very similarly, while 90° would mean they are perpendicular.

Using a Graphing Calculator

A graphing calculator is an incredible tool for performing complex calculations that involve vectors. When finding the angle between vectors, it's useful for quickly obtaining results through built-in functions.
To determine the angle between two vectors, you often calculate the inverse cosine (also known as arccos) function:

• \[ \theta = \cos^{-1}\left(\frac{9}{\sqrt{610}}\right) \approx 81.1^\circ \]

By rounding, we find the angle to be approximately 81°, showing what technology can do fast.These calculators can handle input of vector components and directly compute expressions for cosine and magnitude. They save time from manual computation, which is useful in exams or quick learning sessions. Remember, consistent practice with these tools helps in developing familiarity and speed for problem-solving.