Problem 26 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

Unit vector same direction: \( \langle -0.6, 0.8 \rangle \); opposite direction: \( \langle 0.6, -0.8 \rangle \).

Step 1: Determine the Magnitude of Vector a

To find a unit vector, we need to first calculate the magnitude of \( \mathbf{a} \). The magnitude \( ||\mathbf{a}|| \) can be found using the formula \( ||\mathbf{a}|| = \sqrt{(-3)^2 + 4^2} \). Simplifying, we have \( ||\mathbf{a}|| = \sqrt{9 + 16} = \sqrt{25} = 5 \).

Step 2: Find the Unit Vector in the Same Direction

A unit vector in the same direction as \( \mathbf{a} \) is obtained by dividing each component of \( \mathbf{a} \) by its magnitude. Thus, \( \mathbf{u} = \left\langle \frac{-3}{5}, \frac{4}{5} \right\rangle \). So, \( \mathbf{u} = \left\langle -0.6, 0.8 \right\rangle \).

Step 3: Find the Unit Vector in the Opposite Direction

To find the unit vector in the opposite direction, we take the negative of components of the unit vector in the same direction. Thus, the unit vector opposite to \( \mathbf{a} \) is \( -\mathbf{u} = \left\langle \frac{3}{5}, \frac{-4}{5} \right\rangle \). Therefore, it is \( \left\langle 0.6, -0.8 \right\rangle \).

Key Concepts

Vector Magnitude

The magnitude of a vector provides us with its length or size without regard to its direction. It is calculated using the components of the vector. In the context of a two-dimensional vector, like \( \mathbf{a} = \langle -3, 4 \rangle \), the magnitude is the length of the diagonal of the rectangle formed by these components. This can be calculated using the Pythagorean theorem.

Here's how you compute it:

• Square each component of the vector: \((-3)^2 = 9\) and \(4^2 = 16\).
• Add these squared values together: \(9 + 16 = 25\).
• Take the square root of the sum: \(\sqrt{25} = 5\).

Thus, the magnitude of the vector \( \mathbf{a} \) is 5. This value gives a sense of how large the vector is, regardless of its direction.

Direction of a Vector

The direction of a vector tells us where it points in space. For a vector \( \mathbf{a} = \langle -3, 4 \rangle \), you can visualize its direction as pointing towards the indicated coordinates in a two-dimensional plane.

To express a vector in a particular direction with a specific magnitude, we often look for a unit vector. A unit vector has a magnitude of 1 and points in the same direction as the original vector. You can find this by dividing each component of the vector by its magnitude:

• For \( \mathbf{a} \), divide each component by 5: \(\frac{-3}{5}\) and \(\frac{4}{5}\).
• This results in the unit vector \( \mathbf{u} = \langle -0.6, 0.8 \rangle \).

The unit vector \( \mathbf{u} \) has the same direction as \( \mathbf{a} \) but with a controlled length, making it easy to handle mathematically in various operations.

Opposite Direction Vector

Sometimes, we might need a vector pointing in the exact opposite direction. This is useful when we're interested in reversing the effect or the direction of a vector in space.

The opposite direction vector of any given vector is simply the negative of that vector. For \( \mathbf{a} \), which has a unit counterpart \( \mathbf{u} = \langle -0.6, 0.8 \rangle \), the opposite unit vector is calculated by reversing the signs of its components:

• Take each component of \( \mathbf{u} \) and multiply by -1: \( 0.6 \) and \(-0.8\).
• The opposite direction unit vector is \( \langle 0.6, -0.8 \rangle \).

This simple step efficiently allows us to change the direction of a vector without altering its magnitude, demonstrating the versatility and importance of vector operations in spatial analysis.