Problem 29 - Algebra and Trigonometry

Question

In Exercises \(21-38,\) let $$\mathbf{u}=2 \mathbf{i}-5 \mathbf{j}, \mathbf{v}=-3 \mathbf{i}+7 \mathbf{j}, \text { and } \mathbf{w}=-\mathbf{i}-6 \mathbf{j}$$ Find each specified vector or scalar. $$-4 \mathbf{w}$$

Step-by-Step Solution

Answer

Scaled vector \( -4 \mathbf{w} = 4\mathbf{i} + 24 \mathbf{j} \).

Step 1: Identify the given vector

The vector we are interested in is \( \mathbf{w}=-\mathbf{i}-6 \mathbf{j} \)

Step 2 : Scaling the vector

Scaling of a vector is performed by multiplying each component of the vector by the scalar. Given the scalar as -4, the scaled vector becomes \( -4 \mathbf{w} = -4(-\mathbf{i} - 6 \mathbf{j}) = 4\mathbf{i} + 24 \mathbf{j} \)

Key Concepts

Scaling Vectors

When we talk about scaling vectors, what we mean is adjusting the size, or magnitude, of the vector without changing its direction. This is done by multiplying the vector by a scalar, which is simply a number that stretches or shrinks the vector. This operation keeps the vector's direction the same unless the scalar is negative, which flips the direction.
For example, if you have a vector \( \mathbf{w} = -\mathbf{i} - 6 \mathbf{j} \) and you scale it by \(-4\), you perform the operation on each of its components. This means you multiply each part of the vector by \(-4\), resulting in \( -4 \cdot (-\mathbf{i}) - 4 \cdot 6 \mathbf{j} = 4\mathbf{i} + 24 \mathbf{j} \). The length of the vector changes, but the components are scaled proportionally.

Vector Components

Vectors are described using components that represent its influence along each axis (x, y, or z in 3D). For instance, if you have \( \mathbf{w} = -\mathbf{i} - 6 \mathbf{j} \), its components are based on the standard unit vectors \( \mathbf{i} \) and \( \mathbf{j} \).
The component \( -\mathbf{i} \) indicates a unit movement in the negative x-direction, while \( -6\mathbf{j} \) signifies six units in the negative y-direction. This means \( \mathbf{w} \) is pointing diagonally based on how much each component affects the direction. Understanding components allows us to reconstruct or manipulate vectors easily by applying operations such as addition, subtraction, or scaling.

Scalar Multiplication

Scalar multiplication is the process of multiplying each component of a vector by the same scalar value. This operation directly affects the vector's magnitude and possibly its direction, depending on the sign of the scalar.
A typical experience with scalar multiplication can be seen with \( \mathbf{w} = -\mathbf{i} - 6 \mathbf{j} \) when multiplied by \(-4\). Here, each component will be multiplied individually: \(-4\) with \(-1\) results in \(4\mathbf{i}\), and \(-4\) with \(-6\) gives \(24\mathbf{j}\).

• The magnitude is adjusted by the absolute value of the scalar, stretching the vector if the scalar is greater than 1.
• The sign of the scalar affects the direction: a negative scalar flips the vector's direction.

This way, scalar multiplication acts as a fundamental tool in vector manipulation across fields like physics and engineering.