Problem 15 - Contemporary Precalculus

Question

Find \(u+v, v-u,\) and \(2 u-3 v\). $$\mathbf{u}=2\langle-2,5\rangle, \mathbf{v}=\frac{1}{4}\langle-7,12\rangle$$

Step-by-Step Solution

Answer

Question: Determine the vector sums and differences for the given vectors, where \(\mathbf{u} = 2\langle -2 , 5 \rangle\) and \(\mathbf{v} = \frac{1}{4}\langle -7 , 12 \rangle\). Calculate \(u+v\), \(v-u\), and \(2u-3v\). Answer: The vector sums and differences are: \(u+v = \langle -\frac{23}{4}, 13 \rangle\), \(v-u = \langle \frac{9}{4}, -7 \rangle\), and \(2u-3v = \langle -\frac{53}{4}, 11 \rangle\).

Step 1: Calculate the Vectors

First we need to find the vectors \(\mathbf{u}\) and \(\mathbf{v}\). $$ \mathbf{u} = 2\langle -2 , 5 \rangle = \langle -4 , 10 \rangle $$ $$ \mathbf{v} = \frac{1}{4}\langle -7 , 12 \rangle = \langle -\frac{7}{4} , 3 \rangle $$ Now we have: $$ \mathbf{u} = \langle -4 , 10 \rangle \\ \mathbf{v} = \langle -\frac{7}{4} , 3 \rangle $$

Step 2: Calculate u + v

To find the sum of the vectors \(\mathbf{u}\) and \(\mathbf{v}\), we add their corresponding components together: $$ \mathbf{u} + \mathbf{v} = \langle -4 - \frac{7}{4}, 10 + 3 \rangle = \langle -4 - \frac{7}{4}, 13 \rangle = \langle -\frac{23}{4} , 13 \rangle $$ So the vector sum is: $$ \mathbf{u} + \mathbf{v} = \langle -\frac{23}{4}, 13 \rangle $$

Step 3: Calculate v - u

To find the difference between the vectors \(\mathbf{v}\) and \(\mathbf{u}\), we subtract their corresponding components: $$ \mathbf{v} - \mathbf{u} = \langle -\frac{7}{4} - (-4), 3 - 10 \rangle = \langle \frac{9}{4}, -7 \rangle $$ So the vector difference is: $$ \mathbf{v} - \mathbf{u} = \langle \frac{9}{4}, -7 \rangle $$

Step 4: Calculate 2u - 3v

To find the result of the expression \(2\mathbf{u} - 3\mathbf{v}\), multiply the vectors by the scalars and subtract their corresponding components: $$ 2\mathbf{u} - 3\mathbf{v} = 2 \langle -4 , 10 \rangle - 3 \langle -\frac{7}{4} , 3 \rangle $$ $$ = \langle -8 , 20 \rangle - \langle \frac{21}{4} , 9 \rangle $$ $$ = \langle -8 - \frac{21}{4}, 20 - 9 \rangle $$ $$ = \langle -\frac{53}{4}, 11 \rangle $$ So the result is: $$ 2\mathbf{u} - 3\mathbf{v} = \langle -\frac{53}{4}, 11 \rangle $$

Key Concepts

Vector Addition

Vector addition is a basic operation where two vectors are added together to create a new vector. Imagine vectors as arrows, where the length of the arrow represents the magnitude (how much) and the arrowhead shows the direction. In a coordinate plane, vectors have components usually denoted as \( \langle x , y \rangle \), where \( x \) and \( y \) are the magnitudes along the \( x \)-axis and \( y \)-axis respectively.
To add two vectors, like \( \mathbf{u} = \langle -4, 10 \rangle \) and \( \mathbf{v} = \langle -\frac{7}{4}, 3 \rangle \), simply perform the following steps:

• Add the corresponding components: the first components together and the second components together.
• For the first component, add \( -4 \) and \( -\frac{7}{4} \), resulting in \( -\frac{23}{4} \).
• For the second component, add \( 10 \) and \( 3 \), resulting in \( 13 \).
• Thus, \( \mathbf{u} + \mathbf{v} = \langle -\frac{23}{4}, 13 \rangle \).

By visualizing vectors on a plane, vector addition can resemble placing the tail of one arrow at the head of the other. This new arrow represents the resultant vector.

Vector Subtraction

Vector subtraction is a method that finds the difference between two vectors by considering the negative of one vector and adding it to the other. The operation involves subtracting corresponding components in vector notation.
Consider, for instance, vectors \( \mathbf{v} = \langle -\frac{7}{4}, 3 \rangle \) and \( \mathbf{u} = \langle -4, 10 \rangle \), if you want to calculate \( \mathbf{v} - \mathbf{u} \), follow these steps:

• To find the difference in the first components, subtract \( -4 \) from \( -\frac{7}{4} \), which becomes \( \frac{9}{4} \).
• Subtract the second component of \( \mathbf{u} \) from \( \mathbf{v} \): \( 3 - 10 \) which equals \( -7 \).
• Hence, the resulting vector is \( \langle \frac{9}{4}, -7 \rangle \).

Vector subtraction can be envisioned as adding the opposite of a vector, effectively flipping one vector to point in the opposite direction, before adding it to the other.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number), affecting the magnitude of the vector but not its direction unless the scalar is negative. It is performed by multiplying each component of the vector by the scalar.
If we examine \( 2\mathbf{u} - 3\mathbf{v} \), where \( \mathbf{u} = \langle -4, 10 \rangle \) and \( \mathbf{v} = \langle -\frac{7}{4}, 3 \rangle \), here’s how it unfolds:

• Multiply each component of \( \mathbf{u} \) by 2, resulting in \( \langle -8, 20 \rangle \).
• Multiply each component of \( \mathbf{v} \) by 3, yielding \( \langle \frac{21}{4}, 9 \rangle \).
• Subtract the resulting vector of \( 3\mathbf{v} \) from \( 2\mathbf{u} \) which involves subtracting the corresponding components.
• The first component is \( -8 - \frac{21}{4} = -\frac{53}{4} \).
• The second component is \( 20 - 9 = 11 \).
• The resultant vector is \( \langle -\frac{53}{4}, 11 \rangle \).

Through scalar multiplication followed by addition or subtraction, the vector's size is adjusted without changing its inherent direction unless negatively multiplied, which reverses it.