Problem 33 - Advanced Engineering Mathematics

Question

Find a vector in the opposite direction of \(\mathbf{a}=\langle 4,10\rangle\) but \(\frac{3}{4}\) as long.

Step-by-Step Solution

Answer

The vector is \(\langle -3, -\frac{15}{2} \rangle\).

Step 1: Understand the Opposite Direction

To find a vector in the opposite direction of \(\mathbf{a}=\langle 4,10 \rangle\), we simply need to negate each component of the vector. This gives us the vector \(-\mathbf{a}=\langle -4, -10 \rangle\).

Step 2: Determine the Magnitude Adjustment Factor

The problem requires the new vector to be \(\frac{3}{4}\) as long as the original vector. This means we need to scale the magnitude of the opposite vector by the factor \(\frac{3}{4}\).

Step 3: Apply the Scaling Factor to the Opposite Vector

To find the final vector, multiply each component of the opposite vector \(\langle -4, -10 \rangle\) by \(\frac{3}{4}\). Thus, the final vector is: \(\left\langle \frac{3}{4}(-4), \frac{3}{4}(-10) \right\rangle = \langle -3, -\frac{15}{2} \rangle\).

Key Concepts

Vector Direction

When working with vectors, direction is as crucial as magnitude. The direction of a vector in two-dimensional space is determined by the order and sign of its components. For instance, consider the vector \(\mathbf{a} = \langle 4, 10 \rangle\). Here, the vector extends in the positive x and y directions, indicating its position in the Cartesian plane.

Finding the opposite direction of a vector is straightforward. Simply negate each component of the vector. By reversing the signs of \(\mathbf{a}\)'s components, we transform it into \(-\mathbf{a} = \langle -4, -10 \rangle\).

This action effectively flips the vector 180 degrees in the plane, keeping its magnitude but altering its path entirely. Remember:

• A positive component becomes negative.
• A negative component becomes positive.

This helps achieve the goal of finding a vector pointing in the exact opposite direction of the original.

Scaling Vectors

Scaling vectors involves changing their magnitude while maintaining their direction. This process is comparable to stretching or shrinking a rubber band along the same path.

To scale a vector, multiply each of its components by a scalar (a single number). This scalar directly influences the vector's length - a value greater than 1 will increase the size, while a value between 0 and 1 will decrease it.

Using our vector \(-\mathbf{a} = \langle -4, -10 \rangle\), if we apply a scaling factor of \(\frac{3}{4}\), we adjust each component:

\[ \left\langle \frac{3}{4}(-4), \frac{3}{4}(-10) \right\rangle = \langle -3, -\frac{15}{2} \rangle \]

This reduces the vector's length but retains the new, opposite direction found earlier.

• Scaling alters only the magnitude, not the direction.
• The operation is consistent across all dimensions and directions.

By simply applying standard multiplication, we can easily scale vectors to fit specific requirements.

Magnitude Adjustment

The magnitude of a vector refers to its size or length. Adjusting it involves rescaling the vector to a desired length without affecting its overall direction (unless specifically altered).

Magnitude is derived using the Pythagorean theorem in two dimensions. For a vector \(\mathbf{a} = \langle 4, 10 \rangle\), the magnitude \(|\mathbf{a}|\) is:
\[ \sqrt{4^2 + 10^2} = \sqrt{16 + 100} = \sqrt{116} \]

When a problem requires adjusting the magnitude by a fraction (e.g., \(\frac{3}{4}\)), you're seeking a new vector that retains this fractional size of the original vector's length. By multiplying through this factor, each component scales accordingly:

For the vector \(-\mathbf{a} = \langle -4, -10 \rangle \), applying \(\frac{3}{4}\), results in:
\[ \left\langle \frac{3}{4}(-4), \frac{3}{4}(-10) \right\rangle = \langle -3, -\frac{15}{2} \rangle \]

This calculation helps ensure that the vector’s length is adjusted properly while retaining its precise direction.

• Magnitude adjustment is vital when needing a vector of specific length.
• The mathematical operation enhances precision and versatility in vector manipulation.

Through these steps, one can adeptly manage vector magnitudes across varying applications.