Problem 26 - Precalculus with Calculus Previews

Question

Sketch the vectors \(2 \mathbf{v}\) and \(-2 \mathbf{v}\). $$ \mathbf{v}=\langle 4,7\rangle $$

Step-by-Step Solution

Answer

The vectors \(2 \mathbf{v}\) and \(-2 \mathbf{v}\) are \(\langle 8, 14 \rangle\) and \(\langle -8, -14 \rangle\), respectively.

Step 1: Understand Vector Multiplication

When multiplying a vector by a scalar, each component of the vector is multiplied by that scalar. Here, we need to multiply the vector \( \mathbf{v} = \langle 4, 7 \rangle \) by the scalars 2 and -2.

Step 2: Calculate \(2 \mathbf{v}\)

Multiply each component of \( \mathbf{v} = \langle 4, 7 \rangle \) by 2: \[ 2 \mathbf{v} = 2 \times \langle 4, 7 \rangle = \langle 2 \times 4, 2 \times 7 \rangle = \langle 8, 14 \rangle \]

Step 3: Calculate \(-2 \mathbf{v}\)

Multiply each component of \( \mathbf{v} = \langle 4, 7 \rangle \) by -2: \[ -2 \mathbf{v} = -2 \times \langle 4, 7 \rangle = \langle -2 \times 4, -2 \times 7 \rangle = \langle -8, -14 \rangle \]

Step 4: Sketch the Vectors

On a coordinate plane, plot the initial vector \( \mathbf{v} = \langle 4, 7 \rangle \), then draw \( 2 \mathbf{v} = \langle 8, 14 \rangle \) as a vector starting from the origin, extending more in the same direction as \( \mathbf{v} \), and \( -2 \mathbf{v} = \langle -8, -14 \rangle \) extending in the opposite direction.

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental concept in vector algebra. It involves multiplying a vector by a scalar (a single number). When you perform scalar multiplication, you take each component of the vector and multiply it by the scalar.

For instance, if we have a vector \( \mathbf{v} = \langle 4, 7 \rangle \), and we multiply it by a scalar of 2, we need to multiply both components of the vector by 2. This results in a new vector: \( 2 \mathbf{v} = \langle 2 \times 4, 2 \times 7 \rangle = \langle 8, 14 \rangle \).

Similarly, multiplying by a negative scalar, like -2, reverses the direction of the vector and scales its size. Hence, \( -2 \mathbf{v} = \langle -2 \times 4, -2 \times 7 \rangle = \langle -8, -14 \rangle \).

• Scalar multiplication scales the magnitude of a vector.
• It can also reverse the vector's direction when using a negative scalar.

Understanding scalar multiplication helps in manipulating vectors for various applications, such as scaling dimensions in physics or adjusting graphics in computer simulations.

Coordinate Plane

The coordinate plane, often called the Cartesian plane, is a two-dimensional plane defined by a horizontal axis (the x-axis) and a vertical axis (the y-axis). We use this plane to visualize mathematical concepts like vectors. Each point in this plane is identified by an ordered pair, \( (x, y) \), which represents its position based on these axes.

When you sketch vectors like \( \mathbf{v} = \langle 4, 7 \rangle \), \( 2 \mathbf{v} = \langle 8, 14 \rangle \), and \( -2 \mathbf{v} = \langle -8, -14 \rangle \), you place them as directed lines starting from the origin (0,0) and extending to their defined point.

• The origin on the coordinate plane is the point (0,0).
• The x-coordinate indicates horizontal distance, while the y-coordinate indicates vertical distance.

Understanding the coordinate plane is crucial as it allows you to visualize and solve problems involving vectors easily. It gives a visual perspective to mathematical calculations, enhancing comprehension.

Vector Scaling

Vector scaling refers to increasing or decreasing the size of a vector through scalar multiplication. It's a form of stretching or shrinking the vector's magnitude without altering its direction, unless a negative scalar is involved, which also reverses its direction.

Suppose we have the vector \( \mathbf{v} = \langle 4, 7 \rangle \). Scaling it by 2 transforms it into \( 2 \mathbf{v} = \langle 8, 14 \rangle \), doubling its length. Scaling it by -2 results in \( -2 \mathbf{v} = \langle -8, -14 \rangle \), again doubling its magnitude but in the opposite direction.

• Positive scaling increases a vector's magnitude in the same direction.
• Negative scaling increases magnitude but inverts direction.

The concept of vector scaling is extremely useful in physics for representing forces, velocities, and other vector quantities. It's also crucial in computer graphics for resizing images or objects while maintaining their original proportions.