Problem 23 - Precalculus

Question

Perform the indicated vector operation, given \(u=(-4,3)\) and \(v=\langle 2,-5\rangle\) $$6(\mathbf{u}-\mathbf{v})$$

Step-by-Step Solution

Answer

The result is the vector \((-36, 48)\).

Step 1: Understand Vector Operations

The operation to be performed here is to subtract vector \( \mathbf{v} \) from vector \( \mathbf{u} \), and then multiply the result by 6. This involves vector subtraction followed by scalar multiplication.

Step 2: Subtract the Vectors

To find \( \mathbf{u} - \mathbf{v} \), subtract the components of \( \mathbf{v} \) from \( \mathbf{u} \): \[ \mathbf{u} - \mathbf{v} = (-4, 3) - (2, -5) = (-4 - 2, 3 - (-5)) = (-6, 8) \]

Step 3: Multiply by the Scalar

Now, multiply the resulting vector \((-6, 8)\) by the scalar 6:\[ 6 \times (-6, 8) = (6 \times -6, 6 \times 8) = (-36, 48) \]

Key Concepts

Vector Subtraction

Vector subtraction is the method of subtracting one vector from another. Imagine you have two vectors, say \( \mathbf{u} = (-4, 3) \) and \( \mathbf{v} = (2, -5) \). To subtract these vectors, you need to subtract each respective component of vector \( \mathbf{v} \) from vector \( \mathbf{u} \).

• First, subtract the first component of \( \mathbf{v} \) from the first component of \( \mathbf{u} \): \(-4 - 2 = -6\).


• Next, subtract the second component of \( \mathbf{v} \) from the second component of \( \mathbf{u} \): \(3 - (-5) = 3 + 5 = 8\).

The result of this operation is the vector \((-6, 8)\). This new vector describes the difference between the initial positions and directions of \( \mathbf{u} \) and \( \mathbf{v} \). By conceptualizing vector subtraction as moving from one vector's endpoint back to another's, like arrows, helps solidify the concept in a geometric context.

Scalar Multiplication

Scalar multiplication is when we multiply a vector by a scalar, which is simply a number. Consider the vector \((-6, 8)\) and the scalar \(6\). To perform scalar multiplication, multiply each component of the vector by the scalar.

• Take the first component \(-6\) and multiply it by \(6\): \(6 \times -6 = -36\).


• Similarly, multiply the second component \(8\) by \(6\): \(6 \times 8 = 48\).

The resulting vector is \((-36, 48)\). Scalar multiplication stretches or compresses the original vector without changing its direction, unless the scalar is negative. If the scalar was less than one, it would make the vector shorter, and if more than one, it would make it longer. Meanwhile, a negative scalar flips the direction.

Vector Addition

Vector addition is a fundamental operation where you combine two vectors to form a new vector. With vectors such as \( \mathbf{a} = (x_1, y_1) \) and \( \mathbf{b} = (x_2, y_2) \), the process is straightforward.

• First, add the first components of each vector: \(x_1 + x_2\).


• Then, add the second components: \(y_1 + y_2\).

So, the result is \((x_1 + x_2, y_1 + y_2)\). This operation can be visualized by placing the tail of vector \( \mathbf{b} \) at the head of vector \( \mathbf{a} \). The new vector (resultant) spans from the tail of \( \mathbf{a} \) to the head of \( \mathbf{b} \). This mirrors the summing of forces in physics, where vectors often represent force magnitudes and directions.