Question

The magnitudes of vectors \(\mathbf{A}\) and \(\mathbf{B}\) are given in the following table, as well as the angle between the vectors. For each, find the magnitude \(R\) of the resultant and the angle that resultant makes with vector \(\mathbf{B}\). $$\begin{array}{cc} & {\text { Magnitudes }} \\ \hline A & B & \text { Angle } \\ \hline 1.85 & 2.06 & 136^{\circ} \\ \hline \end{array}$$

Short answer

Magnitude of the resultant vector (R) is calculated using the Law of Cosines, and the angle with vector B is found using the Sine Rule.

Step-by-step solution

Vector Addition Using the Law of Cosines

First, use the Law of Cosines to calculate the magnitude of the resultant vector R. The formula is given by: \( R = \sqrt{A^2 + B^2 - 2AB\cos(\theta)} \), where \( A \) and \( B \) are the magnitudes of the vectors and \( \theta \) is the angle between them. In this case, \( A = 1.85 \), \( B = 2.06 \) and \( \theta = 136^\circ \). Convert the angle to radians for calculation.

Calculate the Magnitude of R

Substitute the given values and calculate \( R \): \( R = \sqrt{1.85^2 + 2.06^2 - 2 \cdot 1.85 \cdot 2.06 \cdot \cos(136^\circ)} \). You will need to use a calculator to find the numerical value.

Vector Addition Using the Sine Rule for Angles

Once you've obtained the magnitude of \( R \), you can calculate the angle the resultant vector makes with vector \( \mathbf{B} \) using the sine rule: \( \frac{\sin(\theta)}{R} = \frac{\sin(\alpha)}{A} = \frac{\sin(\beta)}{B} \), where \( \alpha \) is the angle the resultant makes with vector \( \mathbf{A} \) and \( \beta \) is the angle the resultant makes with vector \( \mathbf{B} \). As you already have the value of \( R \) and the opposite angle, you can find \( \beta \) or \( \alpha \) next.

Calculate the Angle with Vector B

Using \( \beta \), the angle between \( R \) and \( B \), from the sine law: \( \sin(\beta) = \frac{B \cdot \sin(\theta)}{R} \). Substitute the known values and solve for \( \beta \) using the arcsine function on your calculator. Remember to check that the angle is in the correct range based on the original diagram.

Key concepts

Law of Cosines

The Law of Cosines is a critical theorem that establishes a relationship between the lengths of sides and the cosine of one of the angles in any given triangle. This powerful tool extends to a variety of applications, including vector addition, which is crucial in fields like physics and engineering.

When you have two vectors with a known angle between them, as in our exercise, the Law of Cosines helps to determine the magnitude of the resultant vector. The formula resembles a tweaked version of the Pythagorean theorem for non-right triangles and is given as:
\[ R = \sqrt{A^2 + B^2 - 2AB\cos(\theta)} \],
where \( R \) is the magnitude of the resultant vector, \( A \) and \( B \) are the magnitudes of the original vectors, and \( \theta \) is the angle between them. Our example includes a specific angle of \( 136^\circ \); thus, applying the Law of Cosines gives us the resultant vector's magnitude by substituting the given values.

Resultant Vector Magnitude

Vector addition results in a single vector known as the 'resultant vector,' which effectively combines the effect of two or more vectors. The magnitude of this resultant vector is a direct measure of its 'strength' or how much it can influence an object's movement in a particular direction.

To find the magnitude of a resultant vector when two vectors are at an angle that is not 90 degrees, the Law of Cosines comes into play. You've already learned how it is used; by incorporating the magnitudes of the original vectors and the cosine of the included angle, we arrive at the magnitude of the resultant vector. This calculation is vital for understanding how different forces or displacements combine in various orientations, and it's especially useful in analyzing forces not aligned along the same plane or axis.

Sine Rule for Angles

The Sine Rule, also known as the Law of Sines, is another important principle in trigonometry, valuable for analyzing non-right triangles—much like the Law of Cosines. It enables us to find unknown angles or sides when we have information about the ratios of the sides to the sines of their opposite angles.

Applying this rule to our vector problem aids in determining the direction of the resultant vector relative to one of the original vectors. The general formula of the Sine Rule is:
\[ \frac{\sin(\theta)}{R} = \frac{\sin(\alpha)}{A} = \frac{\sin(\beta)}{B} \],
where \( \theta \) is the given angle between vectors \( A \) and \( B \), and \( \alpha \) and \( \beta \) are the angles the resultant vector makes with vectors \( A \) and \( B \), respectively. Once the magnitude of the resultant vector has been calculated, we can rearrange the formula to solve for the unknown angle, which provides critical insight into the vector's orientation after addition.