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 \(\mathbf{v}=\langle-4,1\rangle\) $$3 x-4 u=v$$

Short answer

Question: Given the vectors \(\mathbf{u}=\langle 2,-3\rangle\) and \(\mathbf{v}=\langle-4,1\rangle\), and the equation \(3 \mathbf{x}-4 \mathbf{u}=\mathbf{v}\), find the unknown vector \(\mathbf{x}=\langle a, b\rangle\). Answer: The unknown vector \(\mathbf{x}=\langle a, b\rangle\) is \(\mathbf{x} = \langle\frac{4}{3},-\frac{11}{3}\rangle\).

Step-by-step solution

STEP 1: Understand the given information and write down the vector equation

The given information consists of two vectors \(\mathbf{u}=\langle 2,-3\rangle\) and \(\mathbf{v}=\langle-4,1\rangle\). The vector equation we need to solve for the unknown vector \(\mathbf{x}=\langle a, b\rangle\) is: $$3 \mathbf{x}-4 \mathbf{u}=\mathbf{v}$$

STEP 2: Add \(4 \mathbf{u}\) to both sides of the equation

To isolate \(\mathbf{x}\), we'll need to get rid of the term \(-4 \mathbf{u}\) from the left side of the equation. We do this by adding \(4 \mathbf{u}\) to both sides of the equation: $$3 \mathbf{x}-4 \mathbf{u} + 4 \mathbf{u}=\mathbf{v}+4 \mathbf{u}$$

STEP 3: Simplify the equation

Now, after adding \(4 \mathbf{u}\), the left side simplifies, and the equation becomes: $$3 \mathbf{x}=\mathbf{v}+4 \mathbf{u}$$

STEP 4: Evaluate the right side of the equation

We need to evaluate the right side of the equation by adding the two given vectors, \(\mathbf{v}\) and \(4 \mathbf{u}\). Start by multiplying vector \(\mathbf{u}\) by the scalar \(4\): $$4 \mathbf{u} = 4 \langle 2,-3\rangle = \langle 8,-12\rangle$$ Now, add the vectors \(\mathbf{v}\) and \(4 \mathbf{u}\): $$\mathbf{v} + 4 \mathbf{u} = \langle -4,1\rangle + \langle 8,-12\rangle = \langle -4+8,1-12\rangle =\langle 4,-11\rangle$$ So the equation becomes: $$3 \mathbf{x}=\langle 4,-11\rangle$$

STEP 5: Solve for the unknown vector \(\mathbf{x}\)

Finally, we need to divide the right side of the equation by the scalar \(3\) to find the unknown vector \(\mathbf{x}\): $$\mathbf{x}=\frac{1}{3}\langle 4, -11\rangle = \langle\frac{4}{3},-\frac{11}{3}\rangle$$ So, the unknown vector \(\mathbf{x}=\langle a, b\rangle\) is given by \(\mathbf{x} = \langle\frac{4}{3},-\frac{11}{3}\rangle\).

Key concepts

Vectors

Vectors are fundamental objects in mathematics and physics. They represent quantities that have both magnitude and direction. A vector in a two-dimensional space can be written as \(\mathbf{v} = \langle x, y \rangle\), where \(x\) and \(y\) are its components along the horizontal and vertical axes, respectively.

• Components: The numbers \(x\) and \(y\) in the vector represent how far the vector moves along each axis.
• Magnitude: The length of the vector can be calculated using the Pythagorean theorem: \(\sqrt{x^2 + y^2}\).
• Direction: The direction is typically given as the angle the vector makes with the positive x-axis.

Understanding vectors is a key part of vector algebra, involving operations like vector addition and scalar multiplication.

Scalar Multiplication

Scalar multiplication is when a vector is multiplied by a scalar (real number). This alters the magnitude of the vector but not its direction, unless the scalar is negative, which also reverses its direction.

• Scaling: When a vector \(\mathbf{v} = \langle x, y \rangle\) is multiplied by a scalar \(c\), the result is \(c\mathbf{v} = \langle cx, cy \rangle\).
• Effects on Length: The magnitude of the vector \(c\mathbf{v}\) is \(|c|\) times the magnitude of \(\mathbf{v}\).
• Example: Using the vector \(\mathbf{u} = \langle 2, -3 \rangle\), multiplying by 4 gives \(4\mathbf{u} = \langle 8, -12 \rangle\).

Scalar multiplication is crucial when balancing vector equations, such as aligning coefficients or simplifying expressions.

Vector Addition

Vector addition involves combining two vectors to form a new vector. This operation is performed component-wise.

• Adding Components: To add vectors \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\), the result is \(\mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2 \rangle\).
• Resulting Vector: The new vector represents a cumulative movement.
• Example: Consider \(\mathbf{v} = \langle -4, 1 \rangle\) and \(4\mathbf{u} = \langle 8, -12 \rangle\), their sum is \(\mathbf{v} + 4\mathbf{u} = \langle 4, -11 \rangle\).

This operation is essential in solving equations where vectors need to be expressed in terms of others.

Unknown Vector

Finding an unknown vector involves solving equations where the vector components are not explicitly known. Such problems often require applying operations like vector addition and scalar multiplication to isolate the unknowns.

• Equation Setup: Often starts with a given equation, like \(3\mathbf{x} - 4\mathbf{u} = \mathbf{v}\).
• Isolating the Vector: Manipulate the equation algebraically to solve for the unknown vector \(\mathbf{x}\).
• Example: Rearrange to \(3\mathbf{x} = \langle 4, -11 \rangle\), then solve \(\mathbf{x} = \frac{1}{3}\langle 4, -11 \rangle\) to find \(\mathbf{x} = \langle \frac{4}{3}, -\frac{11}{3} \rangle\).

Mastery of these techniques ensures successful determination of vectors in complex algebraic contexts.