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The dot product, also known as the scalar product, is a way of multiplying two vectors to result in a scalar (a single number). It has significant applications in finding angles between vectors and determining orthogonality. To compute the dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), we use the formula:
\[ \mathbf{a} \cdot \mathbf{b} = a_1 \cdot b_1 + a_2 \cdot b_2 \]
For example, consider vectors \( \mathbf{a} = \langle 1, 5 \rangle \) and \( \mathbf{b} = \langle -3, -2 \rangle \). Their dot product is:

• Multiply the corresponding components: \((1)(-3) = -3\) and \((5)(-2) = -10\).
• Add the results: \(-3 + (-10) = -13\).

So, \( \mathbf{a} \cdot \mathbf{b} = -13 \). The sign of the dot product can be used to infer information about the angle between the vectors:

• If the dot product is positive, the angle is less than 90°.
• If it is zero, the vectors are orthogonal (90°).
• If it is negative, the angle is greater than 90°.

The magnitude of a vector, simply put, is the length or size of the vector. It is calculated using the Pythagorean theorem for vectors in two-dimensional space. For a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \), the magnitude \( |\mathbf{a}| \) is:
\[ |\mathbf{a}| = \sqrt{a_1^2 + a_2^2} \]
This magnitude formula is crucial when you compare vector lengths or need to normalize vectors.
For instance, for vector \( \mathbf{a} = \langle 1, 5 \rangle \), the magnitude is calculated as:

• Square each component: \(1^2 = 1\) and \(5^2 = 25\).
• Add these squares: \(1 + 25 = 26\).
• Take the square root: \(|\mathbf{a}| = \sqrt{26}\).

Similarly, for vector \( \mathbf{b} = \langle -3, -2 \rangle \), the magnitude is:

• Compute squares: \((-3)^2 = 9\) and \((-2)^2 = 4\).
• Add: \(9 + 4 = 13\).
• Square root: \(|\mathbf{b}| = \sqrt{13}\).

Understanding and computing the magnitude of vectors is a fundamental skill in vector operations.

The cosine of the angle between two vectors plays a key role in understanding their spatial relationship. Using the dot product and magnitudes of the vectors, you can find the cosine of the angle between them with the formula:
\[ \text{cos}(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| \cdot |\mathbf{b}|} \]
Here \( \theta \) is the angle between vectors \( \mathbf{a} \) and \( \mathbf{b} \). This formula shows the relation between the dot product and the magnitudes.
Given vectors \( \mathbf{a} = \langle 1, 5 \rangle \) and \( \mathbf{b} = \langle -3, -2 \rangle \), and their dot product \( -13 \) and magnitudes \( \sqrt{26} \) and \( \sqrt{13} \), we find:

• Substitute into the formula:
  \[ \text{cos}(\theta) = \frac{-13}{\sqrt{26} \times \sqrt{13}} \]
• Multiply the magnitudes:
  \( \sqrt{26} \times \sqrt{13} = \sqrt{338} \).
• So,
  \[ \text{cos}(\theta) = \frac{-13}{\sqrt{338}} \].

Using the inverse cosine function, \( \theta = \cos^{-1}\left(\frac{-13}{\sqrt{338}}\right) \), we determine the exact angle, which in this case is approximately
\( 137^{\circ} \). Understanding the cosine of the angle between vectors is essential for analyzing how they interact spatially.