Problem 17 - 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 \(B\) $$A=6.82 \quad B=4.83$$

Step-by-Step Solution

Answer

The magnitude of the resultant vector is \( \sqrt{6.82^2 + 4.83^2} \approx 8.49 \) units, and the angle it makes with vector B is \( \tan^{-1}(\frac{6.82}{4.83}) \approx 54.64 \) degrees.

Step 1: Calculate the Magnitude of the Resultant Vector

Since the vectors are perpendicular, their resultant vector's magnitude can be found using the Pythagorean theorem, which states that the magnitude of the resultant vector (R) is the square root of the sum of the squares of the magnitudes of the original vectors. The formula is: \( R = \sqrt{A^2 + B^2} \).

Step 2: Substitute the Given Magnitudes into the Formula

Substitute the given values for \( A = 6.82 \) and \( B = 4.83 \) into the formula and calculate the resultant: \( R = \sqrt{6.82^2 + 4.83^2} \).

Step 3: Calculate the Angle with Vector B

The angle \( \theta \) that the resultant makes with vector B can be found using the tangent function: \( \tan(\theta) = \frac{A}{B} \), where \( A \) is the opposite side and \( B \) is the adjacent side to the angle in question. Therefore, \( \theta = \tan^{-1}(\frac{A}{B}) \).

Step 4: Substitute the Magnitudes to Find the Angle

Substitute \( A = 6.82 \) and \( B = 4.83 \) into the formula to find \( \theta \): \( \theta = \tan^{-1}(\frac{6.82}{4.83}) \). Then, use a calculator to find the angle in degrees.

Key Concepts

Understanding the Pythagorean Theorem

The Pythagorean theorem is arguably one of the most famous concepts in mathematics, often remembered by its basic formula, \( a^2 + b^2 = c^2 \). This theorem is applicable only in right-angled triangles, where \( a \) and \( b \) represent the lengths of the triangle's legs, and \( c \) is the length of the hypotenuse—the side opposite the right angle.

When applying this theorem to vector resultant calculation, we consider two perpendicular vectors as the legs of a right-angled triangle. The resultant vector then becomes the hypotenuse. By squaring the magnitudes of both vectors and then taking the square root of their sum, you determine the magnitude of this hypotenuse—the resultant vector. In the provided exercise, using values \( A = 6.82 \) and \( B = 4.83 \) for vectors \( A \) and \( B \), respectively, ensures an accurate calculation of the vector's resultant, which effectively illustrates the practical application of the Pythagorean theorem in physics and engineering.

Analyzing the Angle of Vectors

The angle between vectors is an essential aspect of vector analysis, especially when they interact with one another. In a standard XY coordinate system, the angle of a resultant vector with respect to either axis (or another vector) can give us information about its direction.

For perpendicular vectors \( A \) and \( B \) forming a right-angled triangle, the angle \( \theta \) of the resultant vector with either vector can be found using trigonometric ratios. In our case, the angle that the resultant makes with vector \( B \) is our focus. This angle is crucial for understanding the vector's direction and is commonly measured in degrees or radians. When computing the angle in relation to one of the axes or a reference vector, it helps us to visualize and represent the vector's orientation in a two-dimensional plane.

Applying the Tangent Function

In trigonometry, the tangent function relates the angles of a right triangle to the ratio of its opposite side to its adjacent side. It's represented as \( \tan(\theta) \), where \( \theta \) stands for the angle. This function is particularly handy when we know the sides of the triangle but need to find the angle.

When the angle \( \theta \) that the resultant vector makes with vector \( B \) is sought, we use \( \tan(\theta) = \frac{A}{B} \), with \( A \) as the opposite side to \( \theta \) and \( B \) as the adjacent side. To find \( \theta \) itself, we seek the inverse tangent function, denoted as \( \tan^{-1} \), applied to the ratio \( \frac{A}{B} \).

By substituting the values provided for vectors \( A \) and \( B \) into the tangent function, we can then compute the angle in question. Since calculators typically offer the inverse tangent function, this part of the vector resultant calculation is streamlined and easily executed, provided the input is correct. Remember, the output will be in radians by default in most calculators, so ensure that the mode setting is adjusted for degrees if the exercise requires it.