Problem 39 - Precalculus with Calculus Previews

Question

Find a unit vector (a) in the same direction as \(\mathbf{v}\), and \(\mathbf{( b )}\) in the opposite direction of \(\mathbf{v}\). $$ \mathbf{v}=\langle-3,4\rangle $$

Step-by-Step Solution

Answer

Unit vectors are \( \left\langle -\frac{3}{5}, \frac{4}{5} \right\rangle \) and \( \left\langle \frac{3}{5}, -\frac{4}{5} \right\rangle \).

Step 1: Calculate the Magnitude of \( \mathbf{v} \)

The magnitude of a vector \( \mathbf{v} = \langle x, y \rangle \) is calculated using the formula \( \| \mathbf{v} \| = \sqrt{x^2 + y^2} \). For \( \mathbf{v} = \langle -3, 4 \rangle \), it is \( \| \mathbf{v} \| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \).

Step 2: Find Unit Vector in the Same Direction

A unit vector in the same direction is obtained by dividing each component of the vector \( \mathbf{v} \) by its magnitude. Therefore, the unit vector in the same direction as \( \mathbf{v} \) is \( \mathbf{u}_1 = \left\langle \frac{-3}{5}, \frac{4}{5} \right\rangle \).

Step 3: Find Unit Vector in the Opposite Direction

To find a unit vector in the opposite direction, multiply the unit vector in the same direction by \(-1\). Thus, the unit vector in the opposite direction is \( \mathbf{u}_2 = \left\langle \frac{3}{5}, -\frac{4}{5} \right\rangle \).

Key Concepts

Vector Magnitude

Understanding vector magnitude is crucial as it represents the 'length' or 'size' of a vector in precalculus. To compute the magnitude of a vector \(\mathbf{v} = \langle x, y \rangle\), we use the formula: \(\| \mathbf{v} \| = \sqrt{x^2 + y^2}\). This formula is derived from the Pythagorean theorem, where the vector components \(x\) and \(y\) are like the legs of a right triangle, and the magnitude is the hypotenuse.

For example, given vector \(\mathbf{v} = \langle-3, 4\rangle\), we calculate the magnitude as follows:

• Square each component: \((-3)^2 = 9\) and \(4^2 = 16\).
• Add the squares: \(9 + 16 = 25\).
• Take the square root of the sum to find the magnitude: \(\sqrt{25} = 5\).

The vector's magnitude tells us how long the vector is and is always a non-negative number. This concept lays the foundation for understanding how vectors behave geometrically.

Vector Direction

The direction of a vector indicates the specific angle or trajectory it follows from the origin point. Unlike the magnitude, which just indicates size, direction gives us a complete description of the vector's orientation in space.

To find the unit vector in the same direction as a given vector, we simply divide each of its components by the vector's magnitude. This operation normalizes the vector, scaling it to a length of 1, but retaining its original direction.

For instance, using the vector \(\mathbf{v} = \langle-3, 4\rangle\) with magnitude 5, the unit vector \(\mathbf{u}_1\) is found as:

• Divide \(-3\) by 5, resulting in \(-\frac{3}{5}\).
• Divide 4 by 5, which gives \(\frac{4}{5}\).

Thus, \(\mathbf{u}_1 = \left\langle -\frac{3}{5}, \frac{4}{5} \right\rangle\).

To reverse the direction, simply multiply each component of \(\mathbf{u}_1\) by -1. This changes the direction while maintaining the unit magnitude, resulting in \(\mathbf{u}_2 = \left\langle \frac{3}{5}, -\frac{4}{5} \right\rangle\). Understanding both the direction and how to manipulate it is vital for applications of vectors in physics and engineering.

Precalculus

Precalculus lays the groundwork for calculus and higher mathematics. It encompasses fundamental concepts like algebra and geometry that are crucial for understanding advanced mathematical topics such as vectors.

Vectors in precalculus help bridge the gap between algebra and geometry, providing a geometric representation of both direction and magnitude. The principles of vector magnitude and direction align with various precalculus topics:

• Conceptual Understanding: Vectors enhance spatial reasoning and understanding of geometric transformations.
• Problem Solving: They offer a way to visualize and solve problems related to force, motion, and balancing equations in physics.
• Applications: Mastery of vectors in precalculus helps in fields such as computer graphics, engineering, and more.

In precalculus, students learn not just to perform calculations but also to appreciate how these calculations model real-world situations. By mastering concepts like vector magnitude and direction, students are better prepared for calculus, where they will delve into derivative and integral calculus with a strong foundational understanding of these earlier topics.