91Ó°ÊÓ

The dot product, also known as the scalar product of two vectors, is a fundamental operation in vector mathematics. It combines two vectors to produce a scalar (a single number). The formula for the dot product of vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \) is:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \)

This equation shows that you multiply corresponding components of each vector together and then sum these products.
In our exercise, we computed the dot product between \( \mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} \) and \( \mathbf{b} = -4\mathbf{i} + 2\mathbf{k} \).
We multiplied and added the components like this: \( (3)(-4) + (-1)(0) + (2)(2) = -12 + 0 + 4 = -8 \).
The dot product plays a key role in determining the relationship between vectors, such as orthogonality, angles, and projection.

The magnitude of a vector, often called its length, gives you an idea of the size or extent of the vector in space. It is essentially the distance from the origin to the point represented by the vector.
For a vector \( \mathbf{v} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k} \), the magnitude is calculated using the formula:

• \( \|\mathbf{v}\| = \sqrt{x^2 + y^2 + z^2} \)

This is just an application of the Pythagorean theorem in three dimensions.
In our example:

• For \( \mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k} \), the magnitude was \( \|\mathbf{a}\| = \sqrt{3^2 + (-1)^2 + 2^2} = \sqrt{14} \).
• For \( \mathbf{b} = -4\mathbf{i} + 0\mathbf{j} + 2\mathbf{k} \), the magnitude was \( \|\mathbf{b}\| = \sqrt{(-4)^2 + 0^2 + 2^2} = \sqrt{20} = 2\sqrt{5} \).

These magnitudes are essential when calculating the cosine of the angle between vectors, as they ensure we�re comparing the vectors on equal terms.

The cosine of the angle between two vectors provides insight into their directional relationship. When vectors are positioned in the same direction, the cosine is 1, while vectors pointing in opposite directions have a cosine of -1.
The formula for the cosine of the angle \( \theta \) between vectors \( \mathbf{a} \) and \( \mathbf{b} \) is derived from the dot product and magnitudes of the vectors:

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

In our step-by-step solution, we found:

• \( \cos \theta = \frac{-8}{\sqrt{14} \times 2\sqrt{5}} = \frac{-4}{\sqrt{70}} \approx -0.477 \)

This negative value indicates that the vectors point in generally opposite directions.
To determine the angle itself in radians, we used the arc cosine function: \( \theta = \cos^{-1}(-0.477) \approx 2.07 \) radians.
This method links geometry with trigonometry and aids in understanding vector orientations.