Problem 24 - Precalculus Mathematics for Calculus

Question

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 start at (0,0) to (4,-6), (2,3) to (6,-3), and (-3,5) to (1,-1).

Step 1: Understand the Vector Representation

The vector \( \mathbf{u} = \langle 4, -6 \rangle \) means it has a horizontal component of 4 and a vertical component of -6. This means the vector moves 4 units to the right and 6 units down from its starting point.

Step 2: Vector from Initial Point (0,0)

Start at the point \((0,0)\). From this point, move 4 units to the right to reach \((4,0)\) and then 6 units down to reach \((4,-6)\). Draw an arrow from \((0,0)\) to \((4,-6)\), representing the vector.

Step 3: Vector from Initial Point (2,3)

Start at the point \((2,3)\). From this point, move 4 units to the right to reach \((6,3)\) and then 6 units down to reach \((6,-3)\). Draw an arrow from \((2,3)\) to \((6,-3)\), representing the vector.

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

Start at the point \((-3,5)\). From this point, move 4 units to the right to reach \((1,5)\) and then 6 units down to reach \((1,-1)\). Draw an arrow from \((-3,5)\) to \((1,-1)\), representing the vector.

Step 5: Sketch All Representations

Each arrow drawn in Steps 2, 3, and 4 should be of the same length and direction, showing that they all represent the same vector \( \mathbf{u} = \langle 4, -6 \rangle \) from different initial points. Verify each vector starts at its specified initial point and ends at the calculated terminal point.

Key Concepts

Vector Components

When you have a vector like \( \mathbf{u} = \langle 4, -6 \rangle \), it's made up of two parts: the horizontal component and the vertical component. Think of vector components as the building blocks that define the direction and magnitude of the vector. Here:

• The horizontal component, 4, tells us to move right by 4 units.
• The vertical component, -6, instructs us to move down by 6 units.

These components essentially direct how you should "travel" from the start (initial point) to where the vector points you towards (the terminal point). If you visualize this movement on a grid, the vector takes you on a specific path dictated by these components.

Initial Points

An initial point is where your journey with a vector begins. In math problems and real-world applications, vectors don't just float around; they start somewhere specific. The initial point is crucial because it anchors the vector to a particular spot on your coordinate grid.
When sketching vectors like \( \mathbf{u} = \langle 4, -6 \rangle \) from initial points:

• For \((0,0)\), you move to \((4,-6)\).
• For \((2,3)\), the endpoint becomes \((6,-3)\).
• Starting at \((-3,5)\), the vector leads to \((1,-1)\).

The important part is that no matter where the vector starts, its length and direction do not change. Thus, each vector remains identical in terms of magnitude and direction, differing only by where it begins.

Vector Translation

Vector translation involves changing the "place" of a vector without altering its direction or magnitude. It's like picking up the vector and shifting it exactly as it is to another part of the grid. With translation, you're focusing on maintaining:

• The length of the vector.
• Its angle or direction.
• Consistency across different initial points.

In our example with \( \mathbf{u} = \langle 4, -6 \rangle \), even if we move the vector from \((0,0)\) to another position with initial points like \((2,3)\) or \((-3,5)\), the vector itself doesn’t change. This is a fundamental principle underpinning vector translation—the journey (displacement) is the same, only the starting position varies.