Vector Magnitude Basics

A vector in two dimensions, such as \(\vec{a}\), can be expressed in terms of its components along the x-axis and y-axis. If a vector is given as \(\vec{v} = x \vec{i} + y \vec{j}\), where \(\vec{i}\) and \(\vec{j}\) are unit vectors along the x and y axes respectively, then its magnitude (or length) is calculated using the Pythagorean theorem.

The magnitude of vector \(\vec{v}\), denoted as \(|\vec{v}|\), is given by the formula:

\[ |\vec{v}| = \sqrt{x^2 + y^2} \]

This formula essentially finds the hypotenuse of a right-angled triangle formed by the x and y components of the vector.

Applying Vector Magnitude to Find 'm'

In the given problem, we have the vector \(\vec{a} = m \vec{i} + 16 \vec{j}\). Here, the x-component is \(m\) and the y-component is \(16\).

We are also given that the magnitude of vector \(\vec{a}\) is \(|\vec{a}| = 20\).

Using the magnitude formula, we can set up the equation:

\[ |\vec{a}| = \sqrt{m^2 + 16^2} \]

Substitute the given magnitude value into the equation:

\[ 20 = \sqrt{m^2 + 16^2} \]

Solving for 'm' Step-by-Step

To find the value of 'm', we need to isolate 'm' in the equation. Here are the steps:

1. Square both sides of the equation: This removes the square root.

   \[ 20^2 = \left(\sqrt{m^2 + 16^2}\right)^2 \] \[ 400 = m^2 + 16^2 \]

2. Calculate the square of 16:

   \[ 16^2 = 16 \times 16 = 256 \]

   Substitute this value back into the equation:

   \[ 400 = m^2 + 256 \]

3. Isolate \(m^2\): Subtract 256 from both sides of the equation.

   \[ m^2 = 400 - 256 \] \[ m^2 = 144 \]

4. Take the square root of both sides: This will give the value of 'm'.

   \[ m = \pm\sqrt{144} \] \[ m = \pm 12 \]

Since the options provided only include positive integers, we consider the positive value for 'm'.

The value of m is 12.