Problem 20 - Precalculus

Question

Perform the indicated vector operation, given \(u=(-4,3)\) and \(v=\langle 2,-5\rangle\) $$-2 u$$

Step-by-Step Solution

Answer

The vector \(-2u = (8, -6)\).

Step 1: Understand Scalar Multiplication

Scalar multiplication involves multiplying each component of the vector by a scalar value. The scalar in this case is -2, and we will apply it to the vector \(u = (-4, 3)\).

Step 2: Multiply Each Component of the Vector by the Scalar

Multiply the first component of vector \(u\) by -2: \(-2 \times (-4) = 8\)Multiply the second component of vector \(u\) by -2: \(-2 \times 3 = -6\)

Step 3: Write the New Vector

Combine the results from Step 2 to form the new vector after scalar multiplication:\(-2u = (8, -6)\).

Key Concepts

Scalar Multiplication

Scalar multiplication is an essential operation in vector math where each component of a vector is multiplied by a scalar (a single number). This operation essentially scales the vector. Imagine you have a vector that represents a direction and a magnitude, sort of like an arrow. When you apply scalar multiplication, it's like stretching or shrinking that arrow.

For instance, if a vector is \(u = (-4, 3)\), and you scale it by a scalar, say \(-2\), you multiply each element within the vector by \(-2\). This means multiplying the \(x\) component, which is \(-4\), and the \(y\) component, which is \(3\), both separately by \(-2\). This changes the vector to \( (8, -6) \), indicating the vector has been flipped over and stretched in the opposite direction.

• Multiplication of \(-2\) with \(-4\) results in \(8\)
• Multiplication of \(-2\) with \(3\) results in \(-6\)

This transformation is not just about numbers; it's about imagining the shift in position within a space.

Vectors

Vectors are fundamental objects in mathematics and physics, used to represent quantities that have both magnitude and direction. In 2D space, they are often denoted in the form of coordinates, like \( (x, y) \) or \( \langle x, y \rangle \). This is rooted in their need to indicate a specific position in a plane.

Think about vectors as arrows pointing from one position to another, with their length representing magnitude. The earlier exercise gave us vectors \(u=(-4,3)\) and \(v=\langle 2,-5\rangle \) which you can plot to understand their directions.

• \(u=(-4,3)\): Begins from the origin and ends at the point (-4, 3)
• \(v=\langle 2,-5\rangle \): Starts at the origin and finishes at (2, -5)

Learning to interpret vectors is key in many fields like engineering, graphic design, and navigation, wherever direction and magnitude are crucial.

Mathematical Operations

Mathematical operations are procedures or rules you apply in math to manipulate numbers, equations, or expressions. When working with vectors, there are several key operations to remember: addition, subtraction, and scalar multiplication.

In the exercise, scalar multiplication was the focus, but understanding these operations holistically improves problem-solving skills in vector mathematics.

• Scalar Multiplication: Multiply each vector element by a fixed number (scalar).
• Addition: Add corresponding components of two vectors.
• Subtraction: Subtract corresponding components of one vector from another.

All these operations help in reshaping and transforming vectors within their space. Imagine vector operations as a way to shift points, change directions, or scale their size within a graph or grid, which is central to both theoretical mathematics and practical applications.