Problem 34 - Precalculus : Building Concepts and Connections

Question

Perform each operation, given \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) $$\mathbf{u}=\langle 3,2\rangle, \mathbf{v}=\langle-1,4\rangle, \mathbf{w}=\langle-2,-1\rangle$$ $$-4 u+v$$

Step-by-Step Solution

Answer

The result of the operation \(-4\mathbf{u} + \mathbf{v}\) is the vector \(\langle-13,-4\rangle\).

Step 1: Scalar Multiplication

Firstly, perform the scalar multiplication, which consists of multiplying each component of the vector \(\mathbf{u}\) by the scalar -4. \n \((-4) \cdot \mathbf{u} = (-4) \cdot \langle 3,2\rangle = \langle -4*3,-4*2 \rangle= \langle -12,-8 \rangle\)

Step 2: Vector Addition

Next, perform the vector addition operation. This operation is carried out element-wise, meaning that the corresponding components from each vector are added together. \n \((-4) \mathbf{u} + \mathbf{v} = \langle -12,-8 \rangle + \langle -1,4 \rangle = \langle -12-1,-8+4\rangle =\langle -13,-4\rangle\)

Key Concepts

Understanding Scalar Multiplication

Scalar multiplication is a fundamental concept in vector operations. It involves multiplying a vector by a scalar (a single number), which changes the magnitude of the vector without altering its direction, assuming the scalar is positive. If the scalar is negative, the direction is reversed. For example, when you multiply the vector \( \mathbf{u} = \langle 3, 2 \rangle \) by the scalar \(-4\), each component of the vector is multiplied by \(-4\). This transforms the vector into \( \langle -12, -8 \rangle \).

• The original direction remains the same if the scalar is positive and opposite if negative.
• The magnitude of the vector is scaled by the absolute value of the scalar.

Applying scalar multiplication helps in stretching or shrinking vectors based on the numerical factor, enabling diverse applications in geometry, physics, and engineering.

Exploring Vector Addition

Vector addition is another crucial operation in vector mathematics. This operation involves adding two vectors together, component by component. It is akin to overlaying or combining the effects represented by each vector. As per our example, after obtaining \( (-4) \mathbf{u} = \langle -12, -8 \rangle \), we add it to the vector \( \mathbf{v} = \langle -1, 4 \rangle \).The addition is performed as follows:

• Combine the x-components: \(-12 + (-1) = -13\)
• Combine the y-components: \(-8 + 4 = -4\)

Thus, the resultant vector becomes \( \langle -13, -4 \rangle \). This method allows you to visually combine vectors, crucial in fields like physics to calculate net forces, velocities, and more.

Linear Combinations of Vectors

Linear combinations involve the addition of multiple vectors that have been individually multiplied by scalars. This concept is a building block in vector spaces, enabling the creation of new vectors from existing ones. When combining vectors linearly, you can represent movements or transformations.Suppose we're working with vectors \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \). A linear combination like \(-4 \mathbf{u} + \mathbf{v}\) consists of:

• Multiplying \( \mathbf{u} \) by \(-4\) to stretch and reverse its direction.
• Adding the resulting vector to \( \mathbf{v}\) to find a new position or direction.

This results in a fresh vector that embodies a new pathway or position through the linear interplay of \( \mathbf{u} \) and \( \mathbf{v} \). Linear combinations are essential in problems involving solutions to equations, transformations, and various applications in higher mathematics and physical sciences.