Problem 119 - Algebra and Trigonometry

Question

Exercises \(119-121\) will help you prepare for the material covered in the next section. Find the obtuse angle \(\theta,\) rounded to the nearest tenth of a degree, satisfying. $$ \cos \theta=\frac{3(-1)+(-2)(4)}{| \mathbf{v}\|\mathbf{w}\|} $$ where \(\mathbf{v}=3 \mathbf{i}-2 \mathbf{j}\) and \(\mathbf{w}=-\mathbf{i}+4 \mathbf{j}\)

Step-by-Step Solution

Answer

The obtuse angle \(\theta\) satisfying \(\cos \theta=\frac{3(-1)+(-2)(4)}{| \mathbf{v}\|\mathbf{w}\|}\) where \(\mathbf{v}=3 \mathbf{i}-2 \mathbf{j}\) and $\mathbf{w}=-\mathbf{i}+4 \mathbf{j}$ is \(180 - \theta\) degrees, rounded to the nearest tenth.

Step 1: Calculate the dot product of vectors

The dot product of vectors \(\mathbf{v}\) and \(\mathbf{w}\) is calculated as the sum of the product of the corresponding components of the two vectors. Hence, \( \mathbf{v}.\mathbf{w} = (3)(-1)+(-2)(4) = -3-8 = -11 .

Step 2: Calculate the magnitude of vectors

The magnitude 'a' of a vector \(\mathbf{v}=ai+bj\) is given by \(| \mathbf{v}\| = \sqrt{a^2+b^2}\). Therefore, \(|\mathbf{v}\| = \sqrt{3^2+(-2)^2} = \sqrt{9+4} = \sqrt{13}\) and \(|\mathbf{w}\| = \sqrt{(-1)^2+4^2} = \sqrt{1+16} = \sqrt{17}\).

Step 3: Substitute in the given formula

Placing the values of \( \mathbf{v}.\mathbf{w}, |\mathbf{v}\|, |\mathbf{w}\|\) into the cosine formula \(\cos \theta=\frac{\mathbf{v}.\mathbf{w}}{|\mathbf{v}\| .|\mathbf{w}\|}\), we get \(\cos \theta=\frac{-11}{\sqrt{13} . \sqrt{17}}\).

Step 4: Find the acute angle

By taking the inverse cosine of both sides, the acute angle \( \theta \) is calculated as \(\theta = \cos^{-1} \left(\frac{-11}{\sqrt{13} . \sqrt{17}}\right).\)

Step 5: Find the obtuse angle

Since cosine is negative in the second and third quadrants (and an obtuse angle lies in the second quadrant), the obtuse angle is obtained by subtracting the acute angle from \(180^\circ\). Therefore, the obtuse angle \(= 180- \theta \). The angle should be rounded to the nearest tenth.

Key Concepts

Magnitude of a Vector

The magnitude of a vector is a measure of its length. For any vector represented as \( \mathbf{v} = ai + bj \), where \( a \) and \( b \) are the coefficients of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), the magnitude is calculated using the Pythagorean theorem.
The formula for finding the magnitude of such a vector is:

• \( |\mathbf{v}| = \sqrt{a^2 + b^2} \)

This formula stems from considering the vector as a diagonal in a right triangle, where \( a \) and \( b \) are the lengths of the perpendicular sides. By applying this to our vectors \( \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} \) and \( \mathbf{w} = -\mathbf{i} + 4\mathbf{j} \), we find:

• \( |\mathbf{v}| = \sqrt{3^2 + (-2)^2} = \sqrt{13} \)
• \( |\mathbf{w}| = \sqrt{(-1)^2 + 4^2} = \sqrt{17} \)

These magnitudes give us a scalar value representing the length of each vector in a plane.

Obtuse Angle

An obtuse angle is one that measures more than 90 degrees but less than 180 degrees. In the context of vectors and the dot product, the cosine of the angle between two vectors can help determine the nature of their angle.
If the dot product is negative, the angle between the vectors is obtuse. This is because the cosine of angles greater than 90 degrees is negative.
To calculate the obtuse angle \( \theta \) once you have the cosine value, you need to find the acute angle using the inverse cosine function (detailed later). Then, the obtuse angle is found by subtracting the acute angle from 180 degrees:

• Obtuse angle \( = 180^\circ - \theta \)

This formula gives us the correct angle associated with the situation when dealing with vectors pointing in opposite general directions.

Inverse Cosine

The inverse cosine function, denoted as \( \cos^{-1} \), is used to determine the angle whose cosine value is given. In vector mathematics, it assists in finding the angle between two vectors from the cosine of that angle.
The process works as follows:

• Calculate \( \cos \theta \) using the dot product formula: \( \cos \theta = \frac{\mathbf{v} \cdot \mathbf{w}}{|\mathbf{v}| \cdot |\mathbf{w}|} \).
• This will give you a value between -1 and 1, where negative values indicate an obtuse angle.
• Use \( \cos^{-1} \) to find \( \theta \), the acute angle: \( \theta = \cos^{-1}\left(\frac{-11}{\sqrt{13} \cdot \sqrt{17}}\right) \).

Once the acute angle is known, subtract it from 180 degrees to find the obtuse angle:

• Obtuse angle \( = 180^\circ - \theta \)

Using the inverse cosine is crucial in finding the specific angle required from the available cosine value.