Problem 2 - University Calculus: Early Transcendentals

Question

Let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle-2,5\rangle .\) Find the (a) component form and (b) magnitude (length) of the vector. $$-2 \mathbf{v}$$

Step-by-Step Solution

Answer

Component form: \(\langle 4, -10 \rangle\), Magnitude: \(\sqrt{116}\).

Step 1: Find the Scalar Multiplication

To find \(-2 \mathbf{v}\), multiply each component of \(\mathbf{v}\) by -2. Given \(\mathbf{v} = \langle -2, 5 \rangle\), the scalar multiplication is calculated as follows:\(-2 \times \mathbf{v} = \langle -2 \times (-2), -2 \times 5 \rangle = \langle 4, -10 \rangle\). The component form of \(-2 \mathbf{v}\) is \(\langle 4, -10 \rangle\).

Step 2: Calculate the Magnitude of the Vector

To find the magnitude (or length) of the vector \(-2\mathbf{v}\), use the formula for the magnitude of a vector: \(\lVert \mathbf{a} \rVert = \sqrt{a_1^2 + a_2^2}\). Given \(\mathbf{a} = \langle 4, -10 \rangle\), calculate the magnitude as follows:\(\lVert \langle 4, -10 \rangle \rVert = \sqrt{4^2 + (-10)^2} = \sqrt{16 + 100} = \sqrt{116}\). Therefore, the magnitude of \(-2\mathbf{v}\) is \(\sqrt{116}\).

Key Concepts

Scalar Multiplication

Scalar multiplication in vector operations is straightforward. It involves multiplying each component of a vector by a scalar (a real number). For example, if we have a vector \( \mathbf{v} = \langle -2, 5 \rangle \) and want to find \(-2 \mathbf{v}\), we multiply each component by -2. So, we compute:

• \(-2 \times (-2) = 4\)
• \(-2 \times 5 = -10\)

Thus, the resultant vector after scalar multiplication is \( \langle 4, -10 \rangle \). This process effectively scales or stretches the vector by the scalar, potentially changing its direction entirely if the scalar is negative.

Component Form

The component form of a vector is simply writing the vector as an ordered pair or triplet, depending on its dimensions, showing the magnitude of each of its parts along the axes of the coordinate system.For our example, after performing scalar multiplication, we have the vector \(-2 \mathbf{v} = \langle 4, -10 \rangle\). Here, the component form \(\langle 4, -10\rangle\) indicates:

• The first component of the vector is 4, representing movement or effect along the x-axis.
• The second component is -10, representing movement or effect along the y-axis.

The component form is very useful in calculations as it clearly shows the contributions in each direction. It helps in further operations like adding vectors or finding magnitudes.

Vector Magnitude

A vector's magnitude, often thought of as its length, measures how "long" the vector is with respect to the coordinate system it lies in. To find the magnitude of a vector \( \mathbf{a} = \langle a_1, a_2 \rangle \), we use the formula:\[\lVert \mathbf{a} \rVert = \sqrt{a_1^2 + a_2^2}\]In the exercise, for the vector \(-2 \mathbf{v} = \langle 4, -10 \rangle\), the magnitude is calculated by squaring each component, summing them, and then taking the square root:

• First, compute: \(4^2 = 16\)
• Then, compute: \((-10)^2 = 100\)
• Add them: \(16 + 100 = 116\)
• Finally, take the square root: \(\sqrt{116}\)

The magnitude of \(-2 \mathbf{v}\) is therefore \(\sqrt{116}\). This value represents the distance of the vector from the origin in a coordinate plane, showing how far the vector extends.