Problem 15 - Precalculus Mathematics for Calculus

Question

Find the vectors \(\mathbf{u}+\mathbf{v}, \mathbf{u}-\mathbf{v},\) and \(3 \mathbf{u}-\frac{1}{2} \mathbf{v}\) $$\mathbf{u}=\langle 2,-7,3\rangle, \mathbf{v}=\langle 0,4,-1\rangle$$

Step-by-Step Solution

Answer

\(\mathbf{u} + \mathbf{v} = \langle 2, -3, 2 \rangle\), \(\mathbf{u} - \mathbf{v} = \langle 2, -11, 4 \rangle\), \(3\mathbf{u} - \frac{1}{2}\mathbf{v} = \langle 6, -23, 9.5 \rangle\)."}

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

To find \(\mathbf{u} + \mathbf{v}\), we add the corresponding components of \(\mathbf{u}\) and \(\mathbf{v}\) together: \(\langle 2, -7, 3 \rangle + \langle 0, 4, -1 \rangle\). - Add the first components: \(2 + 0 = 2\)- Add the second components: \(-7 + 4 = -3\)- Add the third components: \(3 + (-1) = 2\)Therefore, \(\mathbf{u} + \mathbf{v} = \langle 2, -3, 2 \rangle\).

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

To find \(\mathbf{u} - \mathbf{v}\), we subtract the corresponding components of \(\mathbf{v}\) from \(\mathbf{u}\): \(\langle 2, -7, 3 \rangle - \langle 0, 4, -1 \rangle\).- Subtract the first components: \(2 - 0 = 2\)- Subtract the second components: \(-7 - 4 = -11\)- Subtract the third components: \(3 - (-1) = 4\)Thus, \(\mathbf{u} - \mathbf{v} = \langle 2, -11, 4 \rangle\).

Step 3: Calculate \(3\mathbf{u} - \frac{1}{2}\mathbf{v}\)

First, we calculate \(3\mathbf{u}\) and \(\frac{1}{2}\mathbf{v}\) separately, and then subtract.For \(3\mathbf{u}\): Multiply each component of \(\mathbf{u}\) by 3:- \(3 \times 2 = 6\)- \(3 \times (-7) = -21\)- \(3 \times 3 = 9\)So, \(3\mathbf{u} = \langle 6, -21, 9 \rangle\).For \(\frac{1}{2}\mathbf{v}\): Multiply each component of \(\mathbf{v}\) by \(\frac{1}{2}\):- \(\frac{1}{2} \times 0 = 0\)- \(\frac{1}{2} \times 4 = 2\)- \(\frac{1}{2} \times (-1) = -0.5\)So, \(\frac{1}{2}\mathbf{v} = \langle 0, 2, -0.5 \rangle\).Now, subtract \(\frac{1}{2}\mathbf{v}\) from \(3\mathbf{u}\):- First components: \(6 - 0 = 6\)- Second components: \(-21 - 2 = -23\)- Third components: \(9 - (-0.5) = 9.5\)Therefore, \(3\mathbf{u} - \frac{1}{2}\mathbf{v} = \langle 6, -23, 9.5 \rangle\).

Key Concepts

Understanding Vector Addition

Vector addition is a fundamental operation where vectors are added together through their respective components. Imagine you have two vectors, \(\mathbf{u}\) and \(\mathbf{v}\), each comprising three components. To find their sum, denoted as \(\mathbf{u} + \mathbf{v}\), you simply add corresponding components from each vector.

For instance, if \(\mathbf{u} = \langle 2, -7, 3 \rangle\) and \(\mathbf{v} = \langle 0, 4, -1 \rangle\), the sum is determined by:

• Adding the first components: \(2 + 0 = 2\)
• Adding the second components: \(-7 + 4 = -3\)
• Adding the third components: \(3 + (-1) = 2\)

This results in the new vector: \(\mathbf{u} + \mathbf{v} = \langle 2, -3, 2 \rangle\).

Vector addition is both commutative and associative, meaning you can add vectors in any order, either with multiple vectors together or separately, and still achieve the same result.

Exploring Vector Subtraction

Vector subtraction represents the process of finding the difference between two vectors by subtracting the components of one from the other. Similar to vector addition, vector subtraction involves component-wise operations.

To subtract vector \(\mathbf{v}\) from vector \(\mathbf{u}\), denoted as \(\mathbf{u} - \mathbf{v}\), you determine:

• Subtracting the first components: \(2 - 0 = 2\)
• Subtracting the second components: \(-7 - 4 = -11\)
• Subtracting the third components: \(3 - (-1) = 4\)

Thus, the resulting vector is: \(\mathbf{u} - \mathbf{v} = \langle 2, -11, 4 \rangle\).

Vector subtraction can be thought of as adding a vector with the opposite direction of the second vector, and like addition, it follows the properties of being associative.

Scalar Multiplication in Vectors

Scalar multiplication involves multiplying each component of a vector by a scalar value. When a scalar is multiplied with a vector, each component of that vector gets proportionally adjusted.

For example, with the vector \(\mathbf{u} = \langle 2, -7, 3 \rangle\) and a scalar of 3, the scalar multiplication \(3\mathbf{u}\) is computed as:

• Multiplying the first component by 3: \(3 \times 2 = 6\)
• Multiplying the second component by 3: \(3 \times (-7) = -21\)
• Multiplying the third component by 3: \(3 \times 3 = 9\)

This yields \(3\mathbf{u} = \langle 6, -21, 9 \rangle\).

For another example with \(\frac{1}{2}\mathbf{v}\), if \(\mathbf{v} = \langle 0, 4, -1 \rangle\), each component is multiplied by \(\frac{1}{2}\):

• First component: \(\frac{1}{2} \times 0 = 0\)
• Second component: \(\frac{1}{2} \times 4 = 2\)
• Third component: \(\frac{1}{2} \times (-1) = -0.5\)

Resulting in \(\frac{1}{2}\mathbf{v} = \langle 0, 2, -0.5 \rangle\). Scalar multiplication scales the vector without changing its direction if the scalar is positive.