Problem 31 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

\(2\mathbf{u} = \langle 4, 14 \rangle\), \(-3\mathbf{v} = \langle -9, -3 \rangle\), \(\mathbf{u} + \mathbf{v} = \langle 5, 8 \rangle\), \(3\mathbf{u} - 4\mathbf{v} = \langle -6, 17 \rangle\).

Step 1: Calculate \(2\mathbf{u}\)

To find \(2\mathbf{u}\), multiply each component of \(\mathbf{u}\) by 2. The vector \(\mathbf{u}\) is given by \(\langle 2, 7 \rangle\). Thus, \(2\mathbf{u} = 2 \times \langle 2, 7 \rangle = \langle 2 \times 2, 2 \times 7 \rangle = \langle 4, 14 \rangle\).

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

To find \(-3\mathbf{v}\), multiply each component of \(\mathbf{v}\) by \(-3\). The vector \(\mathbf{v}\) is given by \(\langle 3, 1 \rangle\). Thus, \(-3\mathbf{v} = -3 \times \langle 3, 1 \rangle = \langle -3 \times 3, -3 \times 1 \rangle = \langle -9, -3 \rangle\).

Step 3: Calculate \(\mathbf{u} + \mathbf{v}\)

To find the sum \(\mathbf{u} + \mathbf{v}\), add the corresponding components of the vectors \(\mathbf{u}\) and \(\mathbf{v}\). This results in \(\langle 2 + 3, 7 + 1 \rangle = \langle 5, 8 \rangle\).

Step 4: Calculate \(3\mathbf{u} - 4\mathbf{v}\)

First, calculate \(3\mathbf{u}\) by multiplying each component of \(\mathbf{u}\) by 3, resulting in \(\langle 6, 21 \rangle\). Then, calculate \(4\mathbf{v}\) by multiplying each component of \(\mathbf{v}\) by 4, resulting in \(\langle 12, 4 \rangle\). Now, subtract \(4\mathbf{v}\) from \(3\mathbf{u}\) by subtracting corresponding components: \(\langle 6, 21 \rangle - \langle 12, 4 \rangle = \langle 6 - 12, 21 - 4 \rangle = \langle -6, 17 \rangle\).

Key Concepts

scalar multiplication

Scalar multiplication involves taking a single number, known as a scalar, and using it to multiply each component of a vector separately. Think of a vector as an arrow with both size and direction, and the scalar changes the length of this arrow.
To multiply a vector by a scalar:

• Multiply the scalar with each component of the vector individually.
• The direction of the vector stays the same unless the scalar is negative, which reverses the direction.

For example, with the vector \(\mathbf{u} = \langle 2, 7 \rangle\), multiplying by 2 means performing the operation: \(2 \times \langle 2, 7 \rangle = \langle 4, 14 \rangle\).
This stretches the vector, doubling its length, while maintaining direction.

vector addition

Vector addition is similar to adding numbers, but it occurs component-wise. To find the sum of two vectors, you add the corresponding components.
For two vectors \(\mathbf{u} = \langle 2, 7 \rangle\) and \(\mathbf{v} = \langle 3, 1 \rangle\), the addition \(\mathbf{u} + \mathbf{v}\) is accomplished by:

• Adding the first components: \(2 + 3 = 5\).
• Adding the second components: \(7 + 1 = 8\).

So, the result is \(\langle 5, 8 \rangle\). This means if you were to draw these vectors on a graph, the end of \(\mathbf{v}\) attaches to the head of \(\mathbf{u}\), forming a new vector from the starting point to the final head of the configuration.

subtraction of vectors

The subtraction of vectors is performed by taking corresponding components and finding the difference. It is just like vector addition, except you subtract the values.
When subtracting \(4\mathbf{v}\) from \(3\mathbf{u}\), follow the steps:

• Find the vector for \(3\mathbf{u}\): Multiply \(\mathbf{u} = \langle 2, 7 \rangle\) by 3 to get \(\langle 6, 21 \rangle\).
• Find the vector for \(4\mathbf{v}\): Multiply \(\mathbf{v} = \langle 3, 1 \rangle\) by 4 to get \(\langle 12, 4 \rangle\).
• Subtract corresponding components: \(6 - 12 = -6\) and \(21 - 4 = 17\).

This results in the vector \(\langle -6, 17 \rangle\). The subtraction is essentially adding a vector in the opposite direction, making it feel closer to a movement along a path.

component-wise operations

Component-wise operations are fundamental in vector arithmetic. These operations involve handling each component of a vector individually.
With vectors \(\mathbf{u}\) and \(\mathbf{v}\), these operations include:

• Adding components to find \(\mathbf{u} + \mathbf{v}\): Each component from \(\mathbf{u}\) is summed with the corresponding component from \(\mathbf{v}\).
• Subtracting components for \(\mathbf{u} - \mathbf{v}\): Each component from \(\mathbf{v}\) is subtracted from the corresponding component in \(\mathbf{u}\).
• Multiplication by scalars in individual components, such as taking \(2\mathbf{u}\) or \(-3\mathbf{v}\).

Each operation respects the components as individual units, leading to transformations of the vector as a whole.
Understanding these operations is key to mastering vector arithmetic, as they allow for precise and straightforward manipulation of vectors.