91Ó°ÊÓ

The dot product is a fundamental operation in vector mathematics. It's used to determine the relationship between two vectors. Given vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), the dot product is calculated as: \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \] This results in a scalar, which can reveal important geometric properties, such as:

• The angle between the vectors (a positive dot product indicates an acute angle, a negative indicates obtuse)
• Whether the vectors are perpendicular (when the dot product is zero)

In this exercise, the dot product of vectors \( \mathbf{a} = \langle -5, 12 \rangle \) and \( \mathbf{b} = \langle 4, 6 \rangle \) is 52. This is positive, indicating the vectors point generally in the same direction.

The magnitude of a vector measures its length or size. For a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \), its magnitude is calculated using the formula: \[ \| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2} \] This formula comes from the Pythagorean theorem, applied in the context of vector components.

• The magnitude is always a non-negative quantity.
• It reflects the Euclidean distance from the origin to the point \( \mathbf{a} \).

For our given vector \( \mathbf{a} = \langle -5, 12 \rangle \), the magnitude is calculated as 13. This tells us the vector extends 13 units away from the origin.

The scalar projection of one vector onto another is like casting a shadow. It represents the length of this shadow on the direction of the other vector. The scalar projection of \( \mathbf{b} \) onto \( \mathbf{a} \) is given by: \[ \text{Scalar projection} = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \|} \] This formula balances the dot product with the magnitude of the vector \( \mathbf{a} \), producing a scalar that describes the component of \( \mathbf{b} \) in the direction of \( \mathbf{a} \).

• A positive result indicates a similar direction.
• A negative indicates an opposite direction.

In this example, the scalar projection of \( \mathbf{b} \) onto \( \mathbf{a} \) is 4, indicating they align in direction.

The vector projection of \( \mathbf{b} \) onto \( \mathbf{a} \) provides a vector that reflects how much of \( \mathbf{b} \) is in the direction of \( \mathbf{a} \). Calculated as: \[ \text{Vector projection} = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \|^2} \mathbf{a} \]This formula multiplies the unit vector in the direction of \( \mathbf{a} \) by the scalar projection, hence offering both magnitude and direction.

• It generates a new vector aligned with \( \mathbf{a} \).
• Its magnitude is the scalar projection.

For vectors \( \mathbf{a} = \langle -5, 12 \rangle \) and \( \mathbf{b} = \langle 4, 6 \rangle \), the vector projection is \( \left\langle -\frac{260}{169}, \frac{624}{169} \right\rangle \), showing direction and extent of \( \mathbf{b} \) on \( \mathbf{a} \).