Problem 38 - 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

(a) \(\left\langle -\frac{3}{5}, \frac{4}{5} \right\rangle\); (b) \(\left\langle \frac{3}{5}, -\frac{4}{5} \right\rangle\).

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

The first step in finding a unit vector is to calculate the magnitude of the given vector \(\mathbf{v}\). The formula for the magnitude \(||\mathbf{v}||\) of a vector \(\mathbf{v} = \langle a, b \rangle\) is \(||\mathbf{v}|| = \sqrt{a^2 + b^2}\). For vector \(\mathbf{v} = \langle -3, 4 \rangle\), this yields: \[ ||\mathbf{v}|| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \]

Step 2: Find the Unit Vector in the Same Direction

To find a unit vector in the same direction as \(\mathbf{v}\), divide each component of \(\mathbf{v}\) by its magnitude. Thus, the unit vector \(\mathbf{u}\) is given by: \[ \mathbf{u} = \left\langle \frac{-3}{5}, \frac{4}{5} \right\rangle. \]

Step 3: Find the Unit Vector in the Opposite Direction

The unit vector in the opposite direction can be found by negating the unit vector \(\mathbf{u}\). Therefore, the unit vector in the opposite direction \(\mathbf{w}\) is: \[ \mathbf{w} = \left\langle -\frac{-3}{5}, -\frac{4}{5} \right\rangle = \left\langle \frac{3}{5}, -\frac{4}{5} \right\rangle. \]

Key Concepts

Magnitude of a Vector

The magnitude of a vector is an important concept in vector mathematics. It is like the length or size of the vector. To calculate the magnitude of a vector \( \mathbf{v} = \langle a, b \rangle \), use the formula: \[ ||\mathbf{v}|| = \sqrt{a^2 + b^2}. \]This formula resembles the Pythagorean theorem and effectively measures the vector’s distance from the origin in a coordinate plane.For example, if you have vector \( \mathbf{v} = \langle -3, 4 \rangle \), then its magnitude is:\[ ||\mathbf{v}|| = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. \] By calculating this, you obtain a scalar value that gives an idea of how "long" the vector is. Understanding the magnitude is crucial as it allows us to express vectors in new forms, such as unit vectors.

Vector Direction

Determining the direction of a vector is about finding where it points in space. Knowing the direction is essential to understand what a vector represents beyond just its magnitude. After you have calculated the magnitude of a vector, you can find unit vectors, which are vectors that have a magnitude of 1 but that point in the same or opposite direction as the original vector.To find the unit vector in the same direction as \( \mathbf{v} \), you take each component of the vector and divide it by the magnitude:\[ \mathbf{u} = \left\langle \frac{-3}{5}, \frac{4}{5} \right\rangle. \]

• The direction is maintained as we only adjust the scale of the vector.
• This transformation helps in standardizing the representation of vectors, no matter what unit or scale of the original vector is.

By doing this, the new vector still points in the original direction but is shrunk or "grown" to a consistent length of 1.

Opposite Direction Unit Vector

Sometimes, we need a vector that points exactly in the opposite direction than the original vector. This concept is useful in physics and everyday applications where opposites or reverses are needed. To get a unit vector pointing in the opposite direction, simply negate each component of the unit vector that lies in the same direction:\[ \mathbf{w} = \left\langle -\frac{-3}{5}, -\frac{4}{5} \right\rangle = \left\langle \frac{3}{5}, -\frac{4}{5} \right\rangle. \]

• This flipping of signs ensures the vector points exactly opposite, while maintaining the unit magnitude of 1.
• Opposite direction unit vectors can be used to model scenarios such as force, velocity, or direction, where a reverse arrangement is necessary.

These negations create a unit vector that completely mirrors the path of \( \mathbf{v} \), but in reverse.