Problem 13 - Precalculus Mathematics for Calculus

Question

Find \((a) u \cdot v\) and \((b)\) the angle between \(u\) and \(v\) to the nearest degree. $$\mathbf{u}=\mathbf{i}+3 \mathbf{j}, \quad \mathbf{v}=4 \mathbf{i}-\mathbf{j}$$

Step-by-Step Solution

Answer

(a) The dot product is 1. (b) The angle is approximately 86 degrees.

Step 1: Calculate the Dot Product

To find the dot product \( u \cdot v \), use the formula: \( u \cdot v = u_1v_1 + u_2v_2 \). For vectors \( u = i + 3j \) and \( v = 4i - j \), this becomes \( u \cdot v = 1\cdot4 + 3\cdot(-1) \). Calculate this: \( 4 - 3 = 1 \). Thus, \( u \cdot v = 1 \).

Step 2: Find the Magnitude of u

Calculate the magnitude of vector \( u = i + 3j \). The magnitude formula is \( \|u\| = \sqrt{u_1^2 + u_2^2} \). Substitute the values: \( \|u\| = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \).

Step 3: Find the Magnitude of v

Calculate the magnitude of vector \( v = 4i - j \). The magnitude formula is \( \|v\| = \sqrt{v_1^2 + v_2^2} \). Substitute the values: \( \|v\| = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \).

Step 4: Calculate the Cosine of the Angle

The cosine of the angle \( \theta \) between \( u \) and \( v \) is given by the formula: \( \cos \theta = \frac{u \cdot v}{\|u\| \|v\|} \). Using previous results, \( \cos \theta = \frac{1}{\sqrt{10} \cdot \sqrt{17}} \). Simplify to get \( \cos \theta = \frac{1}{\sqrt{170}} \).

Step 5: Calculate the Angle

To find the angle \( \theta \) in degrees, use the inverse cosine function: \( \theta = \cos^{-1}\left( \frac{1}{\sqrt{170}} \right) \). Calculate this using a calculator and round to the nearest degree. The result is approximately \( 86^{\circ} \).

Key Concepts

Angle Between Vectors

When it's time to figure out the angle between two vectors, we're essentially measuring how far apart they are directionally. You might wonder why this is important. Well, it helps us understand whether the vectors point in similar or very different directions. Here is how we do it:

• First, use the dot product. This product combines the components of the vectors without worrying about their lengths. In our exercise, we found the dot product of vectors \( u \) and \( v \) to be 1.
• Then, find the magnitudes of each vector. This gives us their lengths, acting like a measure of their size.
• With these two pieces of information, we use the formula \( \cos \theta = \frac{u \cdot v}{\|u\| \|v\|} \), which gives us the cosine of the angle \( \theta \) between the vectors. In simple terms, this equation measures how much one vector is aligned with another.

Plugging our values into the formula, the cosine was found to be approximately \( \frac{1}{\sqrt{170}} \), which corresponds to an angle of around \( 86^{\circ} \). Understanding the angle between vectors helps determine their relative orientation.

Magnitude of a Vector

The magnitude of a vector is like its length or size in space. It's important because it helps to grasp how large or small the vector is compared to others. Calculating it is straightforward.

• The magnitude is determined by using the formula \( \|u\| = \sqrt{u_1^2 + u_2^2} \) for a vector \( u \) with components \( (u_1, u_2) \).
• For example, for the vector \( u = i + 3j \), the magnitude was computed as \( \sqrt{10} \).
• This calculation involves squaring each component of the vector, adding these squared values, and finally, taking the square root of the sum.

Knowing the magnitude gives you an insight into how much space the vector covers. It's like figuring out the length of an arrow laid out in a 2D or 3D space.

Inverse Cosine Function

The inverse cosine function, often noted as \( \cos^{-1} \), is crucial when finding angles between vectors. It allows you to retrieve the actual angle once you know the cosine of that angle.

• In our context, once we used the dot product and magnitudes to find the cosine value, which was \( \frac{1}{\sqrt{170}} \), we needed to find the angle \( \theta \). That's where the inverse cosine function steps in.
• By applying \( \theta = \cos^{-1}(x) \), you convert the cosine value \( x \) back into an angle. This angle is usually in radians first unless you specify degrees or convert it.
• In practice, you'd often use a calculator or a software tool to perform this function because it's complex to do manually. In our exercise, the angle between vectors was computed to be approximately \( 86^{\circ} \).

Utilizing the inverse cosine helps give meaningful, easily understood angles from otherwise abstract cosine values.