Problem 56 - Algebra and Trigonometry with Analytic Geometry

Question

Find a vector of magnitude 4 that has the opposite direction of \(a=\langle 2,-5\rangle\).

Step-by-Step Solution

Answer

The vector is \( \left\langle -\frac{8}{\sqrt{29}}, \frac{20}{\sqrt{29}} \right\rangle \).

Step 1: Find the Unit Vector in the Opposite Direction of \( a \)

Start by finding the magnitude of the given vector \( a = \langle 2, -5 \rangle \) using the formula \( \|a\| = \sqrt{2^2 + (-5)^2} \). Calculating, \( \|a\| = \sqrt{4 + 25} = \sqrt{29} \). The unit vector in the same direction as \( a \) is \( \frac{a}{\|a\|} = \left\langle \frac{2}{\sqrt{29}}, \frac{-5}{\sqrt{29}} \right\rangle \). To find the unit vector in the opposite direction, we negate this vector: \( -\left\langle \frac{2}{\sqrt{29}}, \frac{-5}{\sqrt{29}} \right\rangle = \left\langle -\frac{2}{\sqrt{29}}, \frac{5}{\sqrt{29}} \right\rangle \).

Step 2: Scale the Unit Vector to the Desired Magnitude

The unit vector in the opposite direction is \( \langle -\frac{2}{\sqrt{29}}, \frac{5}{\sqrt{29}} \rangle \). To scale this vector to have a magnitude of 4, multiply it by 4: \( 4 \times \left\langle -\frac{2}{\sqrt{29}}, \frac{5}{\sqrt{29}} \right\rangle = \left\langle -\frac{8}{\sqrt{29}}, \frac{20}{\sqrt{29}} \right\rangle \).

Step 3: Simplify the Solution

Simplify the scaled vector: \( \left\langle -\frac{8}{\sqrt{29}}, \frac{20}{\sqrt{29}} \right\rangle \). Optionally, rationalize the denominators if needed to fit specific formats.

Key Concepts

Unit Vector

A unit vector is a vector with a magnitude of exactly 1. It essentially points in the same direction as the original vector but is scaled down to have no length dimension beyond 1.
For any vector, you can find a unit vector by dividing the vector by its magnitude. It is calculated by the formula \[ \hat{u} = \frac{a}{\|a\|} \]Here, \( \hat{u} \) represents the unit vector, \( a \) is the original vector, and \( \|a\| \) is the magnitude of \( a \).
The unit vector effectively gives us a direction indicator, without regard to how strong or weak that direction is except having a unit strength of 1.

• It helps in preserving direction while reducing magnitude.
• Unit vectors are often employed in physics and engineering to help with direction-focused calculations.

Understanding this concept is crucial when you want to change the direction of a vector or when working with vector-oriented mathematics.

Opposite Direction

Vectors pointing in opposite directions are scaled by a factor of -1. When you negate the components of a vector, it flips its direction 180 degrees. This is called finding a vector in the opposite direction.
For a vector \( a = \langle x, y \rangle \), the vector in the opposite direction is given by\[ -a = \langle -x, -y \rangle \]
This action can be particularly useful in physics for representing forces or velocities acting in the opposite sense. For example, if a force is pushing in one direction, the inverse could represent resistance or counterforce. The idea is to maintain the magnitude while altering the orientation completely.

• Opposite vectors are essential in balancing equations and scenarios.
• They demonstrate how direction works in bi-dimensional and tri-dimensional spaces.

Recognizing vectors in opposite directions ensures that you know how to reach or express balance and resistance effectively in vector operations.

Vector Scaling

Vector scaling involves multiplying a vector by a scalar. This operation increases or decreases a vector’s magnitude based on the scalar value without altering its direction. If our goal is to maintain direction while achieving a target magnitude, scaling will be the method used.
If the scalar is:

• Greater than 1, the vector's magnitude increases.
• Less than 1 but greater than 0, the vector's magnitude decreases.
• Negative, the vector's magnitude changes direction and scales by the absolute value of the scalar.

The formula for scaling a vector \( a = \langle x, y \rangle \) by a scalar \( c \) is\[ s = c \times a = \langle c \cdot x, c \cdot y \rangle \]
In practical applications, scaling is pivotal when setting certain conditions like force length in physics, or in graphics when adjusting vector sizes visually.

Magnitude Calculation

Magnitude calculation provides the "length" or "size" of a vector. It's an expression of how much the vector extends from its origin point, in scalar terms. In two dimensions, for vector \( a = \langle x, y \rangle \), the magnitude \( \|a\| \) is calculated with the Pythagorean theorem:\[ \|a\| = \sqrt{x^2 + y^2} \]
Although a vector has direction, its magnitude represents that dimensionless component necessary for the length. Calculating magnitude is vital.

• Vectors with greater magnitude indicate longer extent or higher force in a direction.
• Magnitude remains unaffected by rotation, only changes with recalculations.

In contexts such as physics and engineering, where vector quantities like velocity or force are paramount, accurately calculating magnitude ensures that real, quantitative assessments are correctly made.