Question

Let \(\mathbf{u}=\langle 3,-4\rangle, \mathbf{v}=\langle 1,1\rangle,\) and \(\mathbf{w}=\langle 1,0\rangle\) $$\text { Find }|-2 \mathbf{v}|$$

Short answer

Answer: The magnitude of \(-2\mathbf{v}\) is \(\sqrt{8}\).

Step-by-step solution

Find the vector \(-2\mathbf{v}\)

To find the vector \(-2\mathbf{v}\), we need to multiply each component of the vector \(\mathbf{v}\) by \(-2\). The computation is as follows: $$ (-2)\mathbf{v}=(-2)\langle 1,1\rangle=\langle -2(-1), -2(1)\rangle=\langle 2, -2\rangle $$ So the vector \(-2\mathbf{v}\) is \(\langle 2, -2\rangle\).

Find the magnitude of \(-2\mathbf{v}\)

To find the magnitude of \(-2\mathbf{v}\), we use the formula for the magnitude of a vector, which is \(\lVert\mathbf{v}\rVert = \sqrt{v_{1}^2 + v_{2}^2}\), where \(v_{1}\) and \(v_{2}\) are the components of the vector. In this case, \(v_{1}=2\) and \(v_{2}=-2\). The computation for the magnitude is as follows: $$ \lVert-2\mathbf{v}\rVert=\sqrt{(-2\mathbf{v})_{1}^2+(-2\mathbf{v})_{2}^2} = \sqrt{2^2+(-2)^2} = \sqrt{4+4} = \sqrt{8} $$ So the magnitude of \(-2\mathbf{v}\) is \(\sqrt{8}\).

Key concepts

Understanding Vector Operations

Vector operations are essential in mathematics and physics, allowing us to manipulate vectors for a variety of calculations. A vector is a quantity that has both magnitude and direction, often represented as an arrow in space with its length indicating the magnitude.

Common vector operations include:

• Addition: Vectors can be added by adding their corresponding components. For two vectors \(\mathbf{a} = \langle a_1, a_2 \rangle\) and \(\mathbf{b} = \langle b_1, b_2 \rangle\), their sum is \(\mathbf{a} + \mathbf{b} = \langle a_1 + b_1, a_2 + b_2 \rangle\).
• Subtraction: Similar to addition, subtraction involves subtracting the corresponding components of vectors.
• Scalar Multiplication: Multiplying a vector by a scalar (a real number) involves multiplying each component of the vector by the scalar. This operation stretches or compresses the vector, affecting its magnitude but not its direction unless the scalar is negative, which reverses the vector's direction.

Understanding these basic operations is crucial for working with vectors in any dimensional space.

Exploring Scalar Multiplication

Scalar multiplication is a fundamental concept in vector mathematics. It involves multiplying a vector by a scalar, which is a constant value. This operation changes the size of the vector without altering its direction unless the scalar is negative.

Given a vector \(\mathbf{v}\), represented as \(\langle v_1, v_2 \rangle\), and a scalar \(c\), scalar multiplication is calculated as:

• Formula: \( c\mathbf{v} = \langle cv_1, cv_2 \rangle \)
• Example: If we have \(\mathbf{v} = \langle 1, 1 \rangle\) and a scalar \(-2\), the multiplication \(-2\mathbf{v}\) results in \(\langle -2 \times 1, -2 \times 1 \rangle = \langle -2, -2 \rangle\).

In the exercise, we perform scalar multiplication on vector \(\mathbf{v} = \langle 1, 1 \rangle\) by \(-2\), thus obtaining \(-2\mathbf{v} = \langle 2, -2 \rangle\). This new vector has the same direction but is stretched by a factor of 2 in the opposite direction due to the negative scalar.

Applying the Magnitude Formula

The magnitude of a vector is akin to its length, providing a measure of how far the vector extends in space. Calculating magnitude is a key step in many vector operations, helping understand the vector's size regardless of its direction.

For a vector \(\mathbf{v} = \langle v_1, v_2 \rangle\), the magnitude \(\lVert \mathbf{v} \rVert\) is found using the formula:

• Magnitude Formula: \( \lVert \mathbf{v} \rVert = \sqrt{v_1^2 + v_2^2} \)
• Calculation: For the modified vector \(-2\mathbf{v} = \langle 2, -2 \rangle\), the magnitude is \( \sqrt{2^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} \).

Determining the magnitude helps in understanding and comparing the size of vectors across different vector spaces. It's particularly useful for finding distances in physics, engineering, and graphics.