Problem 37 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find (a) \(\mathbf{u}+\mathbf{v}\) (b) \(\mathbf{u}-\mathbf{v},(\mathbf{c})\) 2 \(\mathbf{u}-3 \mathbf{v},\) and (d) \(\frac{1}{2} \mathbf{v}+4 \mathbf{u} .\) Then sketch each resultant vector. $$\mathbf{u}=\langle 4,2\rangle, \mathbf{v}=\langle 8,0\rangle$$

Step-by-Step Solution

Answer

The resultant vectors are (a) \( \langle 12, 2 \rangle \) (b) \( \langle -4, 2 \rangle \) (c) \( \langle -16, 4 \rangle \) (d) \( \langle 20, 8 \rangle \).

Step 1: Find \( \mathbf{u}+\mathbf{v} \)

To find \( \mathbf{u}+\mathbf{v} \), sum the corresponding components of the two vectors: \( \mathbf{u}+\mathbf{v} = \langle 4 + 8, 2 + 0 \rangle = \langle 12, 2 \rangle \)

Step 2: Find \( \mathbf{u}-\mathbf{v} \)

To find \( \mathbf{u}-\mathbf{v} \), subtract the components of \(\mathbf{v}\) from the corresponding components of \(\mathbf{u}\): \( \mathbf{u}-\mathbf{v} = \langle 4 - 8, 2 - 0 \rangle = \langle -4, 2 \rangle \)

Step 3: Find \(2 \mathbf{u}-3 \mathbf{v}\)

To find \(2 \mathbf{u}-3 \mathbf{v}\), first multiply the vectors \(\mathbf{u}\) and \(\mathbf{v}\) by the scalars 2 and 3 respectively, then subtract the new \(\mathbf{v}\) from the new \(\mathbf{u}\): \(2 \mathbf{u}-3 \mathbf{v} = \langle 2*4 - 3*8, 2*2 - 3*0 \rangle = \langle -16, 4 \rangle \)

Step 4: Find \( \frac{1}{2} \mathbf{v}+4 \mathbf{u} \)

To find \( \frac{1}{2} \mathbf{v}+4 \mathbf{u} \), first multiply the vectors \(\mathbf{u}\) and \(\mathbf{v}\) by the scalars 4 and \( \frac{1}{2} \) respectively, then add the new vectors together: \( \frac{1}{2} \mathbf{v}+4 \mathbf{u} = \langle \frac{1}{2}*8 + 4*4, \frac{1}{2}*0 + 4*2 \rangle = \langle 20, 8 \rangle \)

Step 5: Sketch the resultant vectors

Now we know the resultant vectors from the operations. Despite we can't sketch anything here because of the response format, it's recommended to plot a set of axes, start from the origin, mark the points for the vectors and draw arrows from the origin to these points.

Key Concepts

Vector Addition

Vector addition involves combining two vectors to produce what is known as a resultant vector. To add two vectors together, you sum up their corresponding components. This means you add the x-components together and the y-components together. For instance, if you have two vectors

• \( \mathbf{u} = \langle 4, 2 \rangle \)
• \( \mathbf{v} = \langle 8, 0 \rangle \)

The operation would be \( \mathbf{u} + \mathbf{v} = \langle 4 + 8, 2 + 0 \rangle = \langle 12, 2 \rangle \). In this case, we end up with a new vector that effectively sums up the force and direction of the original vectors. Adding vectors is very much like combining directions on a map, where both magnitude and angle contribute to the original path.
Remember to visualize this operation as shifting one vector so that its tail starts where the other vector's head ends.

Vector Subtraction

Vector subtraction is similar to vector addition but involves subtracting the vector components. Essentially, you take the component of one vector and subtract the corresponding component of another vector. For example, consider again

• \( \mathbf{u} = \langle 4, 2 \rangle \)
• \( \mathbf{v} = \langle 8, 0 \rangle \)

The subtraction \( \mathbf{u} - \mathbf{v} = \langle 4 - 8, 2 - 0 \rangle = \langle -4, 2 \rangle \). This can be visualized on a graph by first drawing the second vector in the reverse direction and then applying vector addition. Instead of adding up the travel, you're effectively considering one vector's negative direction. Imagine taking two steps forward and one backward - your net movement is one step forward.

Scalar Multiplication

Scalar multiplication involves multiplying each component of a vector by a scalar value (a single number). This operation stretches or shrinks the vector but does not change its direction unless the scalar is negative. For instance, when you have the vector

• \( \mathbf{u} = \langle 4, 2 \rangle \)

and multiply it by 2, you have \( 2 \mathbf{u} = \langle 2*4, 2*2 \rangle = \langle 8, 4 \rangle \). This forms a vector twice as long, but with the same direction. Similarly, with the vector

• \( \mathbf{v} = \langle 8, 0 \rangle \)

and a scalar \( \frac{1}{2} \), the result \( \frac{1}{2} \mathbf{v} = \langle 4, 0 \rangle \) is half its original length. This shows how scaling affects the magnitude without altering the initial direction. Consider this akin to increasing the length of a hiking trail while keeping the same compass direction.

Resultant Vector

The resultant vector is the ultimate outcome of vector operations such as addition or subtraction. It represents the total effect of two or more vector quantities. For example, in the operations we performed, we found resultant vectors

• from \( \mathbf{u} + \mathbf{v} \), which was \( \langle 12, 2 \rangle \)
• and from \( \mathbf{u} - \mathbf{v} \), which was \( \langle -4, 2 \rangle \)

These resultant vectors can be illustrated by drawing a vector arrow starting from the origin to the point \( \langle 12, 2 \rangle \) or \( \langle -4, 2 \rangle \) on a coordinate plane. The resultant vector is important for showing combined effects, similar to using multiple pushes to move a box in the same or opposite directions. It’s useful to visualize all vectors on a graph to see how they sum up to the resultant.