Problem 40 - 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 \(\mathbf{v}\). $$ \mathbf{v}=\langle 1,-\sqrt{3}\rangle $$

Step-by-Step Solution

Answer

Unit vector in same direction: \(\langle \frac{1}{2}, -\frac{\sqrt{3}}{2} \rangle\); opposite direction: \(\langle -\frac{1}{2}, \frac{\sqrt{3}}{2} \rangle\).

Step 1: Calculate Magnitude of Vector v

To find a unit vector, you start by calculating the magnitude of the given vector \(\mathbf{v} = \langle 1, -\sqrt{3} \rangle\). The formula for the magnitude of a vector \(\mathbf{v} = \langle a, b \rangle\) is \(\| \mathbf{v} \| = \sqrt{a^2 + b^2}\). Thus, calculate \(\| \mathbf{v} \| = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2\).

Step 2: Find Unit Vector in the Same Direction

A unit vector in the same direction as \(\mathbf{v}\) is found by dividing each component of \(\mathbf{v}\) by its magnitude. Therefore, the unit vector \(\mathbf{u}_1\) is \(\mathbf{u}_1 = \left\langle \frac{1}{2}, \frac{-\sqrt{3}}{2} \right\rangle\).

Step 3: Find Unit Vector in the Opposite Direction

To find a unit vector in the opposite direction, take the negative of each component of the unit vector found in Step 2. Thus, \(\mathbf{u}_2 = \left\langle -\frac{1}{2}, \frac{\sqrt{3}}{2} \right\rangle\).

Key Concepts

Vector Magnitude

Understanding the concept of vector magnitude is essential in the study of unit vectors and many other vector operations. The vector magnitude indicates the length or size of the vector without considering its direction. To compute the magnitude, you can use the formula for a vector \(\mathbf{v} = \langle a, b \rangle\), which is given by \( \| \mathbf{v} \| = \sqrt{a^2 + b^2} \).
This formula is similar to using the Pythagorean theorem, where you calculate the hypotenuse of a right triangle formed by the vector components as its legs.

• The magnitude tells you how long the vector is, irrespective of where it points.
• It always returns a non-negative value.

For the example vector \(\mathbf{v} = \langle 1, -\sqrt{3} \rangle\), its magnitude is calculated as 2. The process involves squaring each component, adding the results, and then taking the square root of this sum. This value is crucial for scaling the vector to create a unit vector.

Opposite Direction

When working with vectors, sometimes you need to find a vector that points in the exact opposite direction. This opposite direction vector is called the negative vector because you negate the components of the original vector to obtain it.

• Negating a vector reverses its direction.
• In the process of finding a unit vector, it involves first finding the unit vector in the same direction and then simply taking the negative sign of each component, providing a balanced and mirror-imaged direction.

For instance, given our vector \(\mathbf{v} = \langle 1, -\sqrt{3} \rangle\), we first find its unit vector \(\mathbf{u}_1 = \langle \frac{1}{2}, \frac{-\sqrt{3}}{2} \rangle\), and then we find the opposite direction unit vector as \(\mathbf{u}_2 = \langle -\frac{1}{2}, \frac{\sqrt{3}}{2} \rangle\).

Same Direction

To find a unit vector in the same direction as the given vector, you basically scale down the original vector so its length becomes one while keeping its direction intact. This is achieved by dividing each component of the vector by its magnitude.

• Unit vectors are especially valuable as they have a magnitude of one.
• They retain the same direction as the original vector.

From our example, the vector \(\mathbf{v} = \langle 1, -\sqrt{3} \rangle\) was scaled to find \(\mathbf{u}_1 = \langle \frac{1}{2}, \frac{-\sqrt{3}}{2} \rangle\), which is its unit version. The direction remains the same — what changes is the scale, shrinking the vector to unit length for simplicity in various calculations and applications.