Problem 2 - Thomas Calculus

Question

In Exercises \(1-8,\) let \(\mathbf{u}=\langle 3,-2\rangle\) and \(\mathbf{v}=\langle- 2,5\rangle .\) Find the (a) component form and \((\mathbf{b})\) magnitude (length) of the vector. $$ -2 \mathbf{v} $$

Step-by-Step Solution

Answer

Component form: \( \langle 4, -10 \rangle \); Magnitude: \( 2\sqrt{29} \).

Step 1: Multiply Vector by Scalar

To find the component form of \(-2 \mathbf{v}\), we first multiply each component of \(\mathbf{v} = \langle -2, 5 \rangle\) by the scalar \(-2\). \[ \langle -2, 5 \rangle \rightarrow -2 \cdot \langle -2, 5 \rangle = \langle -2 \times -2, -2 \times 5 \rangle \] So, the component form is \( \langle 4, -10 \rangle \).

Step 2: Calculate the Magnitude of the Vector

Next, we calculate the magnitude (length) of the vector \( \langle 4, -10 \rangle \).To do this, apply the formula for magnitude: \[ \|\langle 4, -10 \rangle \| = \sqrt{(4)^2 + (-10)^2} \]\[ = \sqrt{16 + 100} \] \[ = \sqrt{116} \]Thus, the magnitude is \( \sqrt{116} \), which simplifies to \( 2\sqrt{29} \) when factoring out the perfect square.

Key Concepts

Scalar Multiplication

Scalar multiplication in vector mathematics involves multiplying each component of a vector by a scalar value (which is just a fancy word for a simple number). This operation changes the size of the vector without altering its direction, unless the scalar is negative. If the scalar is negative, it also reverses the direction of the vector.

Take, for example, the vector \( \mathbf{v} = \langle -2, 5 \rangle\). To find \(\-2 \mathbf{v}\), we multiply each component of the vector by the scalar \(-2\). This gives us:

• For the first component: \(-2 \times -2 = 4\)
• For the second component: \(-2 \times 5 = -10\)

This gives the new vector \(\langle 4, -10 \rangle\), showing how scalar multiplication affects both the magnitude and potentially the direction (here, it flips the direction of component-wise influence) of the vector.

Vector Magnitude

The vector magnitude, also known as the length of the vector, gives us an idea of how long or short a vector is. This can be a useful measurement in physics, engineering, and mathematics to understand the extent a vector covers in space. The magnitude of a vector \(\langle x, y \rangle\) can be calculated using the Pythagorean theorem:

\[ \|\langle x, y \rangle \| = \sqrt{x^2 + y^2} \]

In the example of the calculated vector \(\langle 4, -10 \rangle\), we find its magnitude by plugging the numbers into the formula:

• Square each component: \((4)^2 = 16\) and \((-10)^2 = 100\)
• Sum these squares: \(16 + 100 = 116\)
• Take the square root: \(\sqrt{116} = 2\sqrt{29}\)

The magnitude tells us that our vector \(\langle 4, -10 \rangle\) stretches to about \(2\sqrt{29}\) in length, combining both its x and y components.

Vector Components

Understanding vector components is crucial for visualizing how vectors behave in multidimensional space. A vector, like \(\mathbf{v} = \langle x, y \rangle\), is described by its components. These are essentially projections onto the Cartesian axes (typically represented by i and j in a two-dimensional space), which explain the direction and magnitude in each dimension.

• The first number in the angle brackets, usually associated with the x-axis, depicts how far the vector is lying along the horizontal plane.
• The second number, often aligned with the y-axis, shows the vertical reach of the vector.

In aspects of everyday applications such as navigation or physics, understanding these components helps in breaking down forces, velocities, or any other vectorial quantities into simpler parts.

For example, in our given vector \(\langle 4, -10 \rangle\), the x-component is \4\ and the y-component is \-10\. This tells us that the vector extends 4 units to the right and 10 units down, demonstrating how vector components give us a clear direction and depth for spatial understanding.