Problem 12 - Technical Mathematics with Calculus

Question

In the following problems, the magnitudes \(A\) and \(B\) of two perpendicular vectors are given. Find the resultant and the angle that it makes with \(B\) $$A=2.85 \quad B=4.82$$

Step-by-Step Solution

Answer

The magnitude of the resultant vector is approximately 5.60 units, and it makes an angle \( \theta = \arctan\left(\frac{2.85}{4.82}\right) \) with vector B.

Step 1: Determine the Resultant Vector

For two perpendicular vectors, the magnitude of the resultant vector (R) can be found using the Pythagorean theorem. Here, R is the hypotenuse of the right triangle formed by the vectors A and B. The equation is: \( R = \sqrt{A^2 + B^2} \).

Step 2: Calculating the Magnitude of the Resultant

Substitute the given values into the equation from Step 1: \( R = \sqrt{(2.85)^2 + (4.82)^2} = \sqrt{8.1225 + 23.2324} = \sqrt{31.3549} = 5.60 \).

Step 3: Find the Angle with Vector B

Using trigonometry, the angle \( \theta \) that the resultant makes with vector B can be found using tangent, where \( \tan(\theta) = \frac{A}{B} \). So, \( \theta = \arctan\left(\frac{A}{B}\right) \).

Step 4: Calculating the Angle with Vector B

Substitute the given values into the equation from Step 3: \( \theta = \arctan\left(\frac{2.85}{4.82}\right) \). Calculate this using a calculator set to degree mode to find \( \theta \).

Step 5: Determine the Resulting Angle

The angle you calculate will be the angle between resultant vector R and vector B. Make sure the calculator is in the correct mode (degrees or radians) as required.

Key Concepts

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry. It states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This theorem can be written as an equation: \[ c^2 = a^2 + b^2 \.\] \[c\] is the length of the hypotenuse, and \(a\) and \(b\) are the lengths of the triangle's other two sides.

When it comes to vectors, think of each vector's magnitude as a side length in a right-angled triangle. In our exercise, vector magnitudes \(A\) and \(B\) represent the perpendicular sides of the triangle, and the resulting vector's magnitude \(R\) is the hypotenuse. So, following the Pythagorean theorem, we found \[ R = \sqrt{2.85^2 + 4.82^2} \.\] This formula allowed us to solve for the resultant vector's magnitude, which turned out to be 5.60.

Trigonometry in Vector Addition

Trigonometry is an essential branch of mathematics that deals with the relationships between the sides and angles of triangles. It is particularly useful in vector addition, especially when vectors are perpendicular to each other. In our current problem, we use the concept of a tangent, which is a trigonometric function relating the ratio of the opposite side to the adjacent side in a right-angled triangle.

In relation to vectors, suppose you have vector \(A\) pointing upwards and vector \(B\) pointing to the right, forming a right angle. The tangent of the angle \(\theta\) that the resultant vector makes with \(B\) would be the length of \(A\) divided by the length of \(B\): \[ \tan(\theta) = \frac{A}{B} \.\] Trigonometry in vector addition allows us to calculate not just the magnitude of the resultant vector but also its direction relative to the original vectors, which is crucial in understanding the vector's full characteristics.

Calculating Angles with Arctan

The function arctan or inverse tangent is a way of calculating the angle whose tangent is a given number. It is essentially the reverse operation of finding the tangent of an angle. In vector problems like the one we've solved, after finding the magnitudes using the Pythagorean theorem, we often need to find the direction of the resultant vector. This is where arctan comes into play.

To find the angle that the resultant vector makes with vector \(B\), we set our ratio within the arctan function: \[ \theta = \arctan\left(\frac{A}{B}\right) \.\] By inputting the magnitudes of \(A\) and \(B\), we get \(\theta = \arctan\left(\frac{2.85}{4.82}\right)\). Using a calculator, we can compute this value to get the angle in degrees or radians, depending on what is required for your application. Just remember, when performing this computation, the calculator should be in the correct mode to ensure the angle is accurate to the unit of measure being used.