Problem 13 - APEX Calculus

Question

In Exercises 13-16, find the measure of the angle between the two vectors in both radians and degrees. \(\vec{u}=\langle 1,1\rangle, \vec{v}=\langle 1,2\rangle\)

Step-by-Step Solution

Answer

The angle is approximately 0.6435 radians or 36.87 degrees.

Step 1: Find the Dot Product

The dot product of two vectors \(\vec{u}=\langle a_1, a_2 \rangle\) and \(\vec{v}=\langle b_1, b_2 \rangle\) is found using the formula \(\vec{u} \cdot \vec{v} = a_1b_1 + a_2b_2\). Substitute \(a_1=1, a_2=1\) and \(b_1=1, b_2=2\) into the formula: \(1\cdot1 + 1\cdot2 = 1 + 2 = 3\). Thus, the dot product is 3.

Step 2: Find the Magnitudes of the Vectors

The magnitude of a vector \(\vec{u}=\langle a_1, a_2 \rangle\) is calculated as \(\|\vec{u}\| = \sqrt{a_1^2 + a_2^2}\). Apply this formula to \(\vec{u}\): \(\|\vec{u}\| = \sqrt{1^2 + 1^2} = \sqrt{2}\). For \(\vec{v}\), \(\|\vec{v}\| = \sqrt{1^2 + 2^2} = \sqrt{5}\).

Step 3: Calculate the Cosine of the Angle

The cosine of the angle \(\theta\) between vectors \(\vec{u}\) and \(\vec{v}\) is given by \(\cos \theta = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\|\|\vec{v}\|}\). Substitute the dot product and magnitudes: \( \cos \theta = \frac{3}{\sqrt{2}\sqrt{5}} = \frac{3}{\sqrt{10}}\).

Step 4: Solve for the Angle in Radians

Find \(\theta\) by taking the inverse cosine: \(\theta = \cos^{-1} \left(\frac{3}{\sqrt{10}}\right)\). Using a calculator, find \(\theta\). This value is the angle in radians. \(\theta \approx 0.6435\) radians.

Step 5: Convert the Angle to Degrees

To convert the angle from radians to degrees, use the formula: degrees = radians \(\times \frac{180}{\pi}\). Therefore, \(\theta = 0.6435 \times \frac{180}{\pi} \approx 36.87\) degrees.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector calculus, often forming the starting point in problems involving both direction and magnitude. It is a scalar quantity resulting from the operation applied to two vectors.

For vectors \(\vec{u} = \langle a_1, a_2 \rangle\) and \(\vec{v} = \langle b_1, b_2 \rangle\), the dot product \(\vec{u} \cdot \vec{v}\) is computed using the formula:

• \(\vec{u} \cdot \vec{v} = a_1b_1 + a_2b_2\)

It provides a measure of how aligned two vectors are.

If vectors are perfectly aligned, their dot product is at its maximum. If they are perpendicular, the dot product is zero.

Vector Magnitude

A vector's magnitude is a measure of its "length" in space, and it plays a crucial role when exploring the relationships between vectors such as during the calculation of angles.

The magnitude of a vector \(\vec{u} = \langle a_1, a_2 \rangle\) is given by:

• \(\|\vec{u}\| = \sqrt{a_1^2 + a_2^2}\)

This formula stems from the Pythagorean theorem, representing the hypotenuse length of a right triangle formed by the vector’s components.

Magnitude helps in determining how pronounced the vector is in terms of spatial orientation.

Angle Between Vectors

Finding the angle between vectors involves understanding their geometric relationship, and it's a critical task in vector calculus.

The cosine of the angle \(\theta\) between vectors \(\vec{u}\) and \(\vec{v}\) is expressed as:

• \(\cos\theta = \frac{\vec{u} \cdot \vec{v}}{\|\vec{u}\|\|\vec{v}\|}\)

This formula determines how much one vector "projects" onto another.

Subsequently, \(\theta\) is found by taking the inverse cosine (\(\cos^{-1}\)) of the computed value. Using the inverse trigonometric function, we can discover the actual angle in radians. This method allows us to convert the abstract relationship of vectors into a specific, measurable angle.

Radians and Degrees Conversion

Understanding how to convert between radians and degrees is essential to interpreting and communicating angles in various contexts.

The relationship between radians and degrees is given by the equation:

• degrees = radians \(\times \frac{180}{\pi}\)

Radians are a "natural" unit for angle measurement, relying on the radius of a circle, while degrees offer a more familiar and traditional framework.

In many mathematical applications, angles are calculated in radians, but converting these values into degrees might be necessary for clearer communication or practical applications.