Question

Use the properties of vectors to solve the following equations for the unknown vector \(\mathbf{x}=\langle a, b\rangle .\) Let \(\mathbf{u}=\langle 2,-3\rangle\) and \(\mathbf{v}=\langle-4,1\rangle\) $$2 \mathbf{x}+\mathbf{u}=\mathbf{v}$$

Short answer

Question: Given the two vectors \(\mathbf{u} = \langle 2, -3 \rangle\) and \(\mathbf{v} = \langle -4, 1 \rangle\), and the equation \(2 \mathbf{x}+\mathbf{u}=\mathbf{v}\), find the unknown vector \(\mathbf{x}=\langle a, b\rangle\). Answer: \(\mathbf{x}=\langle -3, 2 \rangle\).

Step-by-step solution

Rewrite the equation

We are given the equation: $$2 \mathbf{x}+\mathbf{u}=\mathbf{v}$$ We must rewrite this equation so that it only contains the unknown vector \(\mathbf{x}\) on one side. First, let's subtract \(\mathbf{u}\) from both sides, then divide both sides by the scalar 2: $$\mathbf{x} = \frac{\mathbf{v} - \mathbf{u}}{2}$$

Substitute in values for \(\mathbf{u}\) and \(\mathbf{v}\)

Now that we have our equation solved for the unknown vector \(\mathbf{x}\), let's substitute in the values provided for \(\mathbf{u}\) and \(\mathbf{v}\): $$\mathbf{x} = \frac{\langle -4, 1 \rangle - \langle 2, -3 \rangle}{2}$$

Calculate the difference of the two vectors

Now we must subtract the two vectors: $$\mathbf{x} = \frac{\langle -4 - 2, 1 - (-3) \rangle}{2}$$ $$\mathbf{x} = \frac{\langle -6, 4 \rangle}{2}$$

Divide by the scalar

We have the numerator of our expression, so now we just need to divide our vector by the scalar 2: $$\mathbf{x} = \langle -3, 2 \rangle$$

Write the solution

Now we know the components of the \(\mathbf{x}\) vector, so we can write the final solution: The unknown vector \(\mathbf{x} = \langle a, b \rangle = \langle -3, 2 \rangle\).

Key concepts

Vector Operations

Vector operations are mathematical procedures that can be performed on vectors, allowing us to manipulate them in various ways. Vectors are often represented as arrows in space or as coordinates, especially in physics and engineering. Here are some fundamental operations that you can perform with vectors:

• **Addition and Subtraction:** These operations involve adding or subtracting corresponding components of the vectors. For example, if you have vector \( \mathbf{a} = \langle x_1, y_1 \rangle \) and vector \( \mathbf{b} = \langle x_2, y_2 \rangle \), the sum is \( \mathbf{a} + \mathbf{b} = \langle x_1 + x_2, y_1 + y_2 \rangle \).


• **Scalar Multiplication:** This involves multiplying each component of a vector by a scalar. For instance, multiplying vector \( \mathbf{a} = \langle x, y \rangle \) by a scalar \( c \) gives \( c\mathbf{a} = \langle cx, cy \rangle \).

These operations are essential in manipulating vectors based on given problems and tasks, such as finding resultant vectors, performing transformations, and solving vector equations.

Scalar Division

Scalar division in vector operations is quite similar to scalar multiplication, but instead of multiplying, you divide each component of the vector by the scalar. It is an essential operation when you need to scale down the vector or determine the unit vector.

In our problem, we scaled the vector \( \langle -6, 4 \rangle\) by dividing by the scalar 2. This operation transforms the vector to \( \langle -3, 2\rangle \). Here’s how you do it:

• Take each component of the vector and individually divide them by the scalar number. For example, if a vector \( \mathbf{v} = \langle x, y \rangle \) is to be divided by a scalar \( c \), the result is \( \frac{\mathbf{v}}{c} = \langle \frac{x}{c}, \frac{y}{c} \rangle \).

Scalar division is vital for determining how the vector's magnitude changes while maintaining the same direction, or vice versa.

Vector Subtraction

Vector subtraction is a fundamental operation where one vector is subtracted from another, resulting in a third vector. This operation is helpful in finding the difference between two vectors or indicating a change from one state to another.

In the exercise, the subtraction operation is shown in how \( \mathbf{u} \) was subtracted from \( \mathbf{v} \):

• Given two vectors \( \mathbf{v} = \langle -4, 1 \rangle \) and \( \mathbf{u} = \langle 2, -3 \rangle \), the subtraction results in \( \mathbf{v} - \mathbf{u} = \langle -4 - 2, 1 + 3 \rangle = \langle -6, 4 \rangle \).

To subtract vectors successfully:

• Subtract the corresponding components. The first component of \( \mathbf{u} \) is subtracted from the first component of \( \mathbf{v} \), and so on.

Understanding vector subtraction is crucial in navigation, engineering, and physics to solve problems related to displacement, velocity changes, and forces among many other applications.