91Ó°ÊÓ

Vector operations are fundamental in the study of linear algebra and have various applications in physics, engineering, and computer science. Understanding vector operations is crucial in solving problems related to direction and magnitude in multi-dimensional spaces. One common operation is the dot product, also known as the scalar product, which takes two vectors and returns a single number (scalar). This operation is performed by multiplying corresponding components of the two vectors and then summing the products. Mathematically, for two vectors \(\mathbf{a}=(a_1, a_2, ... , a_n)\) and \(\mathbf{b}=(b_1, b_2, ... , b_n)\), the dot product is given by \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + ... + a_nb_n\).

In our textbook exercise, the dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) was calculated by following this method. Other vector operations include vector addition, subtraction, and cross product. Each has its own geometric and algebraic interpretation, which students must grasp to solve more complex problems involving vectors.

The magnitude of a vector, often denoted as \(\|\mathbf{v}\|\), is a measure of its length and is always a non-negative scalar value. For a vector in a coordinate space, the magnitude can be visualized as the straight-line distance from the origin to the point defined by the vector. To calculate it, we use the Pythagorean theorem in multi-dimensional space. Specifically, for a vector \(\mathbf{v}=(v_1, v_2, ... , v_n)\), its magnitude is \(\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}\).

In our exercise, the magnitude squared of vector \(\mathbf{u}\), or \(\|\mathbf{u}\|^2\), is found by squaring each component of \(\mathbf{u}\) and adding them, which simplifies the square root operation. This process is directly related to the dot product of \(\mathbf{u}\) with itself, yielding the result of 41.

Scalar multiplication involves multiplying a vector by a scalar (a single number), which scales the vector's magnitude without affecting its direction if the scalar is positive. For a given scalar \(c\) and vector \(\mathbf{v} = (v_1, v_2, ... , v_n)\), scalar multiplication is defined by \(c\mathbf{v} = (cv_1, cv_2, ... , cv_n)\).

This operation is showcased in parts (d) and (e) of the exercise. In part (d), we multiplied each component of \(\mathbf{v}\) by the previously calculated dot product, which demonstrates how the vector scales when multiplied by a scalar. In part (e), the scalar multiplication principle allows us to simplify the problem by first multiplying the scalar with the dot product, because of the distributive property of scalar and vector multiplication. As a result, the dot product of \(\mathbf{u}\) with \(5\mathbf{v}\) is the same as 5 times the dot product of \(\mathbf{u}\) with \(\mathbf{v}\), leading to a solution of -35.