91Ó°ÊÓ

Unit vectors are basic building blocks in vector mathematics. They have a magnitude of 1 and indicate direction along a specific axis.
In 2D space, you often see two unit vectors: \(\hat{i}\) for the x-axis and \(\hat{j}\) for the y-axis.

• \(\hat{i}\) points in the horizontal direction.
• \(\hat{j}\) points in the vertical direction.

For example, a vector \(\vec{a} = 4\hat{i} + 3\hat{j}\) means it moves 4 units in the x-direction and 3 units in the y-direction. By using unit vectors, describing and manipulating vectors becomes straightforward and systematic.

Magnitude represents the length or size of a vector.
To calculate the magnitude of a vector, you can use the Pythagorean theorem.
For a vector \(\vec{v} = a\hat{i} + b\hat{j}\), its magnitude \(|\vec{v}|\) is computed as follows:
\[|\vec{v}| = \sqrt{a^2 + b^2}\]
In our example, we have a resultant vector \(\vec{a} + \vec{b} = -9 \hat{i} + 10 \hat{j}\).
So, the magnitude is:
\[|\vec{a} + \vec{b}| = \sqrt{(-9)^2 + (10)^2}\]
\[|\vec{a} + \vec{b}| = \sqrt{81 + 100}\]\
\[|\vec{a} + \vec{b}| \approx 13.45 \text{ meters}\]
This is the distance from the origin to the point \((-9, 10)\) in 2D space.

Direction tells us where the vector points relative to a reference axis.
To find the direction of a vector, we use trigonometric functions.
For a vector \(\vec{v} = a \hat{i} + b \hat{j}\), we can determine the angle \(\theta\) relative to the positive x-axis using the tangent function:
\[\theta = \tan^{-1} \left( \frac{b}{a} \right)\]
In the example given, the resultant vector is \(\vec{a} + \vec{b} = -9 \hat{i} + 10 \hat{j}\):
\[\theta = \tan^{-1} \left( \frac{10}{-9} \right)\]
\[\theta \approx -48.4^{\circ}\]
The negative sign indicates the direction is below the x-axis.
To express this in terms of the positive x-axis, we add 180°:
\[\theta = 180^{\circ} - 48.4^{\circ}\]\[\theta \approx 131.6^{\circ}\]
So, our vector points at an angle of approximately 131.6° from the positive x-axis, going counterclockwise.