Problem 7 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-8: Find (a) the dot product of the two vectors and (b) the angle between the two vectors. $$ \langle 10,7\rangle, \quad\left\langle-2,-\frac{7}{5}\right\rangle $$

Step-by-Step Solution

Answer

The dot product is \(-\frac{149}{5}\) and the angle is \(180^{\circ}\).

Step 1: Define the Dot Product Formula

The dot product of two vectors \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\) is given by the formula: \(\mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2\).

Step 2: Substitute the Components

Substitute the vector components from the given vectors \(\mathbf{u} = \langle 10, 7 \rangle\) and \(\mathbf{v} = \langle -2, -\frac{7}{5} \rangle\) into the dot product formula. This gives: \(10 \cdot (-2) + 7 \cdot \left(-\frac{7}{5}\right)\).

Step 3: Calculate the Dot Product

Perform the calculations: \(10 \cdot (-2) = -20\) and \(7 \cdot \left(-\frac{7}{5}\right) = -\frac{49}{5}\). The dot product is \(-20 - \frac{49}{5}\).

Step 4: Combine the Terms

To combine \(-20\) and \(-\frac{49}{5}\), convert \(-20\) to a fraction with a denominator of 5: \(-\frac{100}{5}\). Now, combine: \(-\frac{100}{5} - \frac{49}{5} = -\frac{149}{5}\).

Step 5: Define the Cosine Angle Formula

The angle \(\theta\) between two vectors is found using the formula: \(\cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \cdot \|\mathbf{b}\|}\), where \(\|\mathbf{a}\|\) and \(\|\mathbf{b}\|\) are the magnitudes of the vectors.

Step 6: Compute the Magnitude of Each Vector

The magnitude of a vector \(\mathbf{a} = \langle a_1, a_2 \rangle\) is \(\|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2}\). For \(\mathbf{u} = \langle 10, 7 \rangle\), \(\|\mathbf{u}\| = \sqrt{10^2 + 7^2} = \sqrt{149}\). For \(\mathbf{v} = \langle -2, -\frac{7}{5} \rangle\), \(\|\mathbf{v}\| = \sqrt{(-2)^2 + \left(-\frac{7}{5}\right)^2} = \sqrt{4 + \frac{49}{25}} = \sqrt{\frac{149}{25}} = \frac{\sqrt{149}}{5}\).

Step 7: Plug into Cosine Formula

Substitute the calculated values into the cosine formula: \(\cos \theta = \frac{-\frac{149}{5}}{\sqrt{149} \cdot \frac{\sqrt{149}}{5}} = \frac{-\frac{149}{5}}{\frac{149}{5}} = -1\).

Step 8: Determine the Angle

If \(\cos \theta = -1\), then \(\theta = \pi\) radians or \(180^{\circ}\), meaning the vectors are in opposite directions.

Key Concepts

Angle Between Vectors

The angle between two vectors can tell us a lot about their orientation in space. To find this angle, we often use the dot product, which connects the vectors' magnitudes and the cosine of the angle between them. The relationship is given by the formula:

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

This means the dot product of the vectors divided by the product of their magnitudes gives the cosine of the angle between them. By solving this equation for \(\theta\), you can determine the exact angle. It's quite interesting to note that if the cosine of this angle is -1, the vectors are pointing in completely opposite directions, meaning they form an angle of 180°, or \(\pi\) radians.

Vector Magnitude

The magnitude of a vector is like the length of a vector in a particular space. We calculate it using the Pythagorean Theorem in the context of vector components. For a vector \(\mathbf{a} = \langle a_1, a_2 \rangle\), the formula is:

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

This essentially boils down to finding the hypotenuse of a right triangle formed by the vector's components. This measure plays a crucial role in normalizing vectors and computing dot products. In the given problem, the magnitudes computed demonstrate this.Calculating \(\|\mathbf{u}\| = \sqrt{10^2 + 7^2}\) results in \(\sqrt{149}\), which gives a real number length of the vector \(\mathbf{u}\). Similarly, vector \(\mathbf{v}\) is expressed with its components to find its magnitude, resulting in \(\frac{\sqrt{149}}{5}\), showing how fractions can often arise in these calculations.

Cosine of Angle Between Vectors

The cosine of the angle between two vectors carries significant meaning in vector mathematics. It quantifies the extent to which two vectors are pointing in the same direction.To find the cosine value, we use the following formula:

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

Once the dot product and the magnitudes of the vectors are computed, substituting them into this formula will yield the cosine of the angle. Putting it in context, if \(\cos \theta = 1\), the vectors are perfectly aligned, and if \(\cos \theta = -1\), they are in opposite directions as in the given exercise.This specific result, where \(\cos \theta = -1\), indicates a 180-degree angle between the vectors. It marks a full reversal in directionality, providing a quick but complete understanding of their mutual opposition.

Opposite Vectors

Vectors are said to be opposite if they point in exactly opposite directions. This means the angle between them is \(180^{\circ}\) or \(\pi\) radians. When this happens, their dot product will yield a specific value that corresponds to the cosine of \(-1\). This signals that they are directly opposing each other in orientation. In practical terms, if you were looking at two such vectors on a plane, they would form a straight line, pointing in opposite directions with no inclination towards one another. The calculations in the exercise reveal this using simple mathematics to demonstrate how certain values, like \(\cos \theta = -1\), embody the concept of opposition in vector terms.Opposite vectors are a fundamental concept, especially when resolving forces or looking at directional paths in various fields of science and engineering. Understanding this helps demystify many physical phenomena where forces act in reverse directions.