Problem 9 - Precalculus: Functions and Graphs

Question

Sketch vectors corresponding to \(a, b, a+b\) \(2 a,\) and \(-3 b\) $$\mathbf{a}=\langle- 4,6\rangle, \quad \mathbf{b}=\langle- 2,3\rangle$$

Step-by-Step Solution

Answer

Plot vectors starting from the origin based on the given coordinates and scalars.

Step 1: Plot Vector \(\mathbf{a}\)

To represent the vector \(\mathbf{a} = \langle -4, 6 \rangle\), start at the origin (0,0) on the Cartesian plane. The vector moves left 4 units (since the x-component is -4), and then moves up 6 units (since the y-component is 6). Mark the terminal point of \(\mathbf{a}\) at (-4, 6) and draw an arrow from the origin to this point.

Step 2: Plot Vector \(\mathbf{b}\)

For the vector \(\mathbf{b} = \langle -2, 3 \rangle\), start again at the origin. This vector moves left 2 units and then up 3 units. Mark the terminal point at (-2, 3) and draw an arrow from the origin to this point.

Step 3: Plot Vector \(\mathbf{a} + \mathbf{b}\)

The vector \(\mathbf{a} + \mathbf{b}\) is the sum of \(\mathbf{a}\) and \(\mathbf{b}\). Add the components: \((-4) + (-2) = -6\) and \(6 + 3 = 9\). Start at the origin, move left 6 units, and up 9 units. Mark the terminal point at (-6, 9) and draw an arrow from the origin.

Step 4: Plot Vector \(2\mathbf{a}\)

The vector \(2\mathbf{a}\) is a scalar multiplication of \(\mathbf{a}\). Multiply each component of \(\mathbf{a}\) by 2: \(2 \times -4 = -8\) and \(2 \times 6 = 12\). Start at the origin, move left 8 units, and up 12 units. Mark the terminal point at (-8, 12) and draw an arrow from the origin.

Step 5: Plot Vector \(-3\mathbf{b}\)

The vector \(-3\mathbf{b}\) is another scalar multiplication. Multiply each component of \(\mathbf{b}\) by -3: \(-3 \times -2 = 6\) and \(-3 \times 3 = -9\). Start at the origin, move right 6 units, and down 9 units. Mark the terminal point at (6, -9) and draw an arrow from the origin.

Key Concepts

Vector Addition

Vector addition is the process of combining two or more vectors to produce a resultant vector. This involves adding the corresponding components of the vectors together.

Imagine you have vectors \(\mathbf{a} = \langle -4, 6 \rangle\) and \(\mathbf{b} = \langle -2, 3 \rangle\). To find their sum, \(\mathbf{a} + \mathbf{b}\), you perform addition on each component separately:

• For the x-components: \(-4 + (-2) = -6\).
• For the y-components: \(6 + 3 = 9\).

The resultant vector \(\mathbf{a} + \mathbf{b}\) is therefore \(\langle -6, 9 \rangle\).
This visualizes as shifting the position of one vector to the endpoint of another; their sum forms a straight path from the start to the finish of the combined journey.

Scalar Multiplication

Scalar multiplication involves multiplying each component of a vector by a single number, called a scalar.

This operation changes the magnitude (length) of the vector but not its direction, unless the scalar is negative. When a negative scalar is used, the vector also reverses direction.For example, multiplying \(\mathbf{a} = \langle -4, 6 \rangle\) by 2 gives:

• The x-component becomes: \(2 \times -4 = -8\).
• The y-component becomes: \(2 \times 6 = 12\).

Thus, the vector \(2\mathbf{a}\) is \(\langle -8, 12 \rangle\)
Likewise, \(-3\mathbf{b}\) means applying a -3 scalar to \(\mathbf{b} = \langle -2, 3 \rangle\):

• x-component: \(-3 \times -2 = 6\).
• y-component: \(-3 \times 3 = -9\).

This gives \(-3\mathbf{b} = \langle 6, -9 \rangle\), a vector pointing in the opposite direction to \(\mathbf{b}\) with a greater magnitude.

Component Form

The component form of a vector is a way to express the vector using its horizontal and vertical components.

These components represent how far the vector moves along each axis: the x-component for horizontal movement and the y-component for vertical movement.To write vector \(\mathbf{a} = \langle -4, 6 \rangle\) in component form, interpret it as:

• -4 indicates a move left by 4 units on the x-axis.
• 6 suggests an upward climb of 6 units on the y-axis.

Likewise, vector \(\mathbf{b} = \langle -2, 3 \rangle\) involves:

• Shifting left two units
• And rising by three units.

Component form helps simplify calculations such as addition and scalar multiplication by focusing on the effects along each axis separately.

Plotting Vectors

Plotting vectors on a graph involves marking their magnitudes and directions directly onto a coordinate plane.

To plot a vector such as \(\mathbf{a} = \langle -4, 6 \rangle\), you begin at the origin or any specified starting point. From there:

• Move horizontally by the x-component; in this case, 4 units to the left, resulting in an x-position of -4.
• Next, move vertically by the y-component; here, ascend 6 units, leading to a y-position of 6.

The endpoint is \((-4, 6)\). Draw an arrow from the origin to this point; it visually represents \(\mathbf{a}\).
Similarly, plot \(\mathbf{b} = \langle -2, 3 \rangle\) by moving 2 units left and 3 units up from the origin, ending at \((-2, 3)\).Graphs help show vector relationships like additions or scalar changes, making it easier to visualize vector operations.