Problem 24 - Algebra and Trigonometry

Question

\(23-26\) Sketch representations of the given vector with initial points at \((0,0),(2,3),\) and \((-3,5)\). $$ \mathbf{u}=\langle 4,-6\rangle $$

Step-by-Step Solution

Answer

Vectors end at (4, -6), (6, -3), and (1, -1).

Step 1: Understanding the Problem

The vector \( \mathbf{u} = \langle 4, -6 \rangle \) is to be represented starting from different initial points given in the problem: \((0,0), (2,3),\) and \((-3,5)\). A vector is defined by its direction and magnitude, so we need to find its terminal points from each initial point.

Step 2: Vector from Origin

For the initial point \((0,0)\), the vector is simply represented by the terminal point \( (0+4, 0-6) \). So, the terminal point is \( (4, -6) \).

Step 3: Vector from Point (2,3)

Starting from the initial point \((2,3)\), the vector \( \mathbf{u} \) can be represented as shifting 4 units to the right (x-direction) and 6 units down (y-direction), resulting in the terminal point \( (2+4, 3-6) = (6, -3) \).

Step 4: Vector from Point (-3,5)

From the initial point \((-3, 5)\), the vector is represented by moving 4 units to the right, and 6 units down, giving us the terminal point \( (-3+4, 5-6) = (1, -1) \).

Step 5: Conclusion

We have sketched and identified the terminal points for the vector \( \mathbf{u} = \langle 4, -6 \rangle \) from each of the given start points: \( (4, -6), (6, -3), \) and \( (1, -1) \).

Key Concepts

Initial Point

In vector representation, the initial point is where a vector begins. It serves as the starting coordinate from which the vector's direction and magnitude are applied. In the context of the exercise, you are given three initial points:

• The origin \((0,0)\).
• The point \((2,3)\).
• The point \((-3,5)\).

These initial points act as the base from which the vector \( \mathbf{u} = \langle 4, -6 \rangle \) starts.
The choice of an initial point affects the position of the vector on the coordinate plane, though the vector's inherent properties (direction and magnitude) remain unchanged. Understanding this is crucial for drawing or sketching vectors.
For each starting point, you will calculate the terminal point the vector extends to, defining its position on the plane.

Terminal Point

The terminal point of a vector is the endpoint that you reach when applying the vector's direction and magnitude starting from an initial point. It's often represented as the head of the vector arrow. In our exercise, you calculate the terminal points as follows:

• For \( (0,0) \): Add the vector components to the initial point, resulting in \( (4, -6) \).
• For \( (2,3) \): Move 4 units right and 6 units down, arriving at \( (6, -3) \).
• For \( (-3,5) \): Shift in the same direction and magnitude, ending at \( (1, -1) \).

The terminal point visually indicates where the vector leads. Remember, for accurate sketches or when solving problems involving vectors, calculating the terminal point accurately is key. Always apply the vector components carefully to avoid mistakes.

Magnitude and Direction

A vector's magnitude and direction define its essential properties. The magnitude measures the vector's length, and the direction shows where the vector points. For our vector \( \mathbf{u} = \langle 4, -6 \rangle \), this can be explained as follows:

• The magnitude is calculated using the Pythagorean theorem: \[ \sqrt{4^2 + (-6)^2} = \sqrt{16 + 36} = \sqrt{52} = \sqrt{4 \times 13} \approx 7.21 \text{ units} \]
• The direction is determined by the ratio of its components, and it suggests a move 4 units right (positive x-direction) and 6 units down (negative y-direction).

Understanding both aspects is crucial as they dictate how vectors interact in a vector space. Regardless of the initial point, the magnitude and direction dictate the path and length of the vector.

Coordinate Plane Representation

Representing a vector on a coordinate plane visually illustrates its position using initial and terminal points. Each vector \( \mathbf{u} = \langle 4, -6 \rangle \) starts from its respective initial point and extends to a terminal point:

• From \( (0,0) \) to \( (4, -6) \),
• From \( (2,3) \) to \( (6, -3) \),
• From \( (-3,5) \) to \( (1, -1) \).

This visualization allows for an intuitive grasp of vectors' linear transformation. By plotting initial and terminal points on the grid, you create a clear path that illustrates the vector's components applied in sequence.
Coordinate plane representations are foundational in areas such as physics, engineering, and computer graphics. The clear demonstration of start and stop locations helps in tasks like calculating resultants or modeling motion.