91Ó°ÊÓ

Vector addition is a fundamental operation in mathematics, particularly in precalculus, where vectors are often used to represent quantities that have both magnitude and direction. This operation can be visualized by placing the tail of one vector at the head of another and drawing the resultant vector from the tail of the first to the head of the second vector.

To perform vector addition algebraically, simply add the corresponding components of each vector. For example, suppose we have two vectors in a 2-dimensional space, \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \). The sum of these vectors is given by \( \mathbf{a} + \mathbf{b} = \langle a_1+b_1, a_2+b_2 \rangle \).

In the context of the given exercise, vector addition is used when combining vectors \( \mathbf{v} \) and \( \mathbf{w} \). This is done by adding their respective components, resulting in the new vector \( \langle -3, 3 \rangle \) which represents the combined effect of \( \mathbf{v} \) and \( \mathbf{w} \).

Remember

While adding vectors, keep in mind that the addition is commutative, meaning that \( \mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a} \), as it does not matter in which order the vectors are added.

Scalar multiplication is the process of multiplying a vector by a scalar (a real number), which changes the magnitude of the vector but not its direction, unless the scalar is negative, in which case the direction is reversed.

To multiply a vector by a scalar, you simply multiply each component of the vector by the scalar. For instance, if we have a scalar \( c \) and a vector \( \mathbf{x} = \langle x_1, x_2 \rangle \), then the scalar multiplication is \( c\mathbf{x} = \langle cx_1, cx_2 \rangle \).

In our given exercise, scalar multiplication comes into play when we multiply the vector \( \mathbf{-u} \) by -1. This operation flips the vector in the opposite direction due to the negative scalar, producing a new vector, \( \langle -3, -2 \rangle \) from \( \mathbf{u} \) which initially was \( \langle 3, 2 \rangle \).

Key Point

It is crucial to understand that scalar multiplication affects both the magnitude and the direction (in case of a negative scalar), and the result is still a vector in the same or opposite direction.

The dot product, also called the scalar product, is a way of multiplying two vectors that results in a scalar. This operation measures the magnitude of one vector in the direction of another and is used extensively in physics, engineering, and computer graphics, among other fields.

To calculate the dot product of two vectors, you multiply the corresponding components of the vectors and then sum up these products. Given two 2-dimensional vectors \( \mathbf{p} = \langle p_1, p_2 \rangle \) and \( \mathbf{q} = \langle q_1, q_2 \rangle \), their dot product is \( \mathbf{p} \cdot \mathbf{q} = p_1q_1 + p_2q_2 \).

In our exercise, we are calculating the dot product of vectors \( \mathbf{-u} \) and \( \mathbf{v}+\mathbf{w} \), which requires taking the sum of the products of their respective components, resulting in the scalar value 3. The dot product tells us about the alignment of the two vectors: if the dot product is positive, the vectors are pointing in more or less the same direction; if negative, they point in opposite directions; and if zero, they are orthogonal (at a right angle to each other).

Practical Usage

The dot product can be particularly useful for determining the angle between two vectors, as it is related to the cosine of the angle between them. When the dot product is zero, the vectors are perpendicular, a valuable piece of information in many geometric scenarios.