Problem 33 - 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. $$3 \mathbf{v}-4 \mathbf{w}$$

Step-by-Step Solution

Answer

The calculated vector \(3\mathbf{v}-4\mathbf{w}\) equals to \(-5\mathbf{i}+45\mathbf{j}\)

Step 1: Multiply the vectors by their respective scalars

To start, each vector needs to be multiplied by its given scalar. This is done by multiplying each component of the vector by the scalar, resulting in new vectors.\n\nSo, \( 3\mathbf{v} = 3(-3 \mathbf{i} + 7 \mathbf{j}) = -9\mathbf{i} + 21\mathbf{j} \)\n and \( -4\mathbf{w} = -4(-\mathbf{i} - 6 \mathbf{j}) = 4\mathbf{i} + 24\mathbf{j} \).

Step 2: Add the resulting vectors

The next step is to add the two vectors that resulted from the scalar multiplications. This is done by adding the i components together and the j components together. \n\nSo, \( 3\mathbf{v} - 4\mathbf{w} = (-9\mathbf{i} + 21\mathbf{j}) + (4\mathbf{i}+24\mathbf{j}) = -5\mathbf{i}+45\mathbf{j} \).

Step 3: Express the result in i-j form

Finally, express the result in i-j notation. So, the vector \(3\mathbf{v}-4\mathbf{w}\) equals to \(-5\mathbf{i}+45\mathbf{j}\)

Key Concepts

Vector Addition

Understanding vector addition is crucial when dealing with the complex relationships between multiple vectors in physics and engineering. Imagine you're at a shopping mall with multiple pathways. To reach your favorite store, you need to walk a certain distance forward and then a certain distance right. Each of these pathways can be represented as vectors, and the total path you take is the sum of these two vectors.

When you add two vectors, you're essentially combining their paths to find a new direction and distance you'd travel if you followed one right after the other. When using vectors like \textbf{u}, \textbf{v}, and \textbf{w} in the exercise, you want to combine movements in each direction. For coordinate vectors, this means adding their \(i\) terms together and their \(j\) terms together, as if you were combining footstep counts in the east and north directions to reach a new location on a map.

By following this simple rule, we can quickly determine new paths and even predict paths resulting from various movements or forces.

I-J Notation

If you're new to working with vectors, think of \(i\) and \(j\) as friendly assistants guiding you along the horizontal and vertical paths on a grid. In the world of vectors, these assistants are known as the 'unit vectors.' The \(i\) vector stands for one unit of movement horizontally, while the \(j\) vector represents one unit of movement vertically, reminiscent of playing old video games where you could only move in straight lines up, down, left, or right.

In the i-j notation, every vector in two dimensions can be expressed as a combination of these basic moves. For example, \(2i - 5j\) means 'move two units to the right and five units down.' This notation simplifies complicated movements into easy-to-understand instructions. It allows for simple addition and subtraction of movements, as you would when finding the best path to get around blockages on your route to a store.

Linear Combination of Vectors

In mathematics, a 'linear combination' is a blend of items that keeps things in a straight line. With vectors, it means mixing different paths by stretching or shrinking them (using scalar multiplication) and then piling them onto each other to make a new path. Think of this as taking your shopping paths from earlier: One path goes towards the electronics section, and another towards the book store. By tweaking the length of these paths (say, double the distance to electronics and half towards books), you customize a new route.

In our exercise, we adjust the vectors \(\textbf{v}\) and \(\textbf{w}\) by multiplying them with the scalars 3 and -4. Then we combine them by vector addition. This mixing and matching of paths gives us a precise route in two-dimensional space, akin to giving your friend specific instructions to get to a hidden spot in the mall. It's a nifty way to direct movements, model forces, or predict the outcome of event sequences.