Question

(a) What is the sum in unit-vector notation of the two vectors \(\overrightarrow{\mathbf{a}}=5 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}\) and \(\overrightarrow{\mathbf{b}}=-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}} ?(b)\) What are the magni- tude and the direction of \(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}\) ?

Short answer

The sum of the two vectors in unit-vector notation is \(\overrightarrow{\mathbf{r}} = 2\hat{\mathbf{i}} +5\hat{\mathbf{j}}\). The magnitude of the resultant vector \(\overrightarrow{\mathbf{r}}\) is \(\sqrt{29}\), and its direction is approximately 68.20 degrees.

Step-by-step solution

Vector Addition in Unit-Vector Notation

Add the given vectors component by component. The i components are added separately from the j components. The sum for the i components is \(5 - 3 = 2\), and the sum for the j components is \(3 + 2 = 5\). So the resultant vector \(\overrightarrow{\mathbf{r}}= \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}} = 2\hat{\mathbf{i}} + 5\hat{\mathbf{j}} \) .

Calculating the Magnitude of the Resultant Vector

The magnitude (length) of a vector in unit-vector notation is given by \(\sqrt{(i_{component})^{2} + (j_{component})^{2}}\). The magnitude of vector \(\overrightarrow{\mathbf{r}}\) is \(\sqrt{(2)^{2} + (5)^{2}} = \sqrt{29}\).

Finding the Direction of the Resultant Vector

The direction (angle) of a vector in unit-vector notation in the xy-plane is given by the formula \(\theta = \arctan(\frac{j_{component}}{i_{component}})\). The direction of vector \(\overrightarrow{\mathbf{r}}\) is \(\arctan(\frac{5}{2})\). Using a calculator, this comes out to approximately 68.20 degrees (in quadrants I and II).

Key concepts

Unit-Vector Notation

Unit-vector notation is a way of expressing vectors using unit vectors as building blocks. A unit vector has a magnitude of one and points in a specific direction. In two dimensions, we use \( \hat{\mathbf{i}} \) for the x-direction and \( \hat{\mathbf{j}} \) for the y-direction. By combining these unit vectors, we can describe any vector in the plane.
For example, the vector \( \overrightarrow{\mathbf{a}} = 5 \hat{\mathbf{i}} + 3 \hat{\mathbf{j}} \) means the vector has 5 units in the x direction and 3 units in the y direction. Similarly, \( \overrightarrow{\mathbf{b}} = -3 \hat{\mathbf{i}} + 2 \hat{\mathbf{j}} \) tells us this vector goes 3 units in the negative x direction and 2 units in the positive y direction.

• Unit vectors are essential for breaking down vectors into their components.
• This notation simplifies addition, subtraction, and other operations.

To add vectors in unit-vector notation, simply add their respective components, as seen in the step-by-step solution above.

Magnitude of a Vector

The magnitude of a vector refers to its length, which can be found using the Pythagorean theorem when the vector components are known. For a vector expressed in unit-vector notation as \( \overrightarrow{\mathbf{r}} = a \hat{\mathbf{i}} + b \hat{\mathbf{j}} \), its magnitude \( |\overrightarrow{\mathbf{r}}| \) is calculated as \( \sqrt{a^2 + b^2} \).
In the example provided, the resultant vector \( \overrightarrow{\mathbf{r}} = 2 \hat{\mathbf{i}} + 5 \hat{\mathbf{j}} \) has a magnitude \( \sqrt{2^2 + 5^2} = \sqrt{29} \).

• The magnitude tells us how long the vector is, regardless of direction.
• This concept is important in physics, engineering, and everyday applications.

Remember, while the magnitude gives us the 'size', it does not inform us of the vector's direction.

Direction of a Vector

The direction of a vector indicates where the vector is pointing within the coordinate system and is commonly measured as an angle from a standard direction, such as the positive x-axis. This angle, often represented by \( \theta \), can be found using the tangent function when the vector components are known.
For a vector \( \overrightarrow{\mathbf{r}} = a \hat{\mathbf{i}} + b \hat{\mathbf{j}} \), the direction is determined by \( \theta = \arctan\left(\frac{b}{a}\right) \).

• This formula arises from trigonometry, specifically from the right triangle formed by the vector’s components.
• In our example, \( \overrightarrow{\mathbf{r}} = 2 \hat{\mathbf{i}} + 5 \hat{\mathbf{j}} \), the direction is \( \arctan\left(\frac{5}{2}\right) \), roughly 68.20 degrees from the positive x-axis.

Knowing the direction is crucial for fully understanding the vector, as it complements the magnitude.