Problem 24 - Precalculus

Question

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

Step-by-Step Solution

Answer

The resulting vector is \((-30, 33)\).

Step 1: Scalar Multiplication of Vector u

First, multiply vector \(u=(-4, 3)\) by 2: \[2u = 2(-4, 3) = (-8, 6)\]

Step 2: Scalar Multiplication of Vector v

Next, multiply vector \(v=\langle 2, -5 \rangle\) by 3: \[3v = 3(2, -5) = (6, -15)\]

Step 3: Calculate 4u

Multiply vector \(u=(-4, 3)\) by 4: \[4u = 4(-4, 3) = (-16, 12)\]

Step 4: Combine Results

Add the results from Step 1, then subtract the result from Step 2, and finally add the result from Step 3: \[(2u + 4u) - 3v = (-8, 6) + (-16, 12) - (6, -15)\]

Step 5: Vector Addition and Subtraction

Combine all the vectors: - Add the results from Step 1 and Step 3: \[(-8, 6) + (-16, 12) = (-24, 18)\] - Now, subtract the result from Step 2: \[(-24, 18) - (6, -15) = (-24 - 6, 18 - (-15)) = (-30, 33)\]

Key Concepts

Scalar Multiplication

Scalar multiplication is a fundamental operation in vector algebra, where a scalar (a real number) is used to scale a vector. When you multiply a vector by a scalar, each component of the vector is multiplied by the scalar.

Here's how it works:

• For a vector \( u = (x, y) \), multiplying by a scalar \( k \) results in \( ku = (kx, ky) \).

If we look at the example given, there are two scalar multiplications:

• First, vector \( u = (-4,3) \) is multiplied by 2: \( 2u = 2(-4, 3) = (-8, 6) \).
• Similarly, vector \( v = \langle 2, -5 \rangle \) is multiplied by 3: \( 3v = 3(2, -5) = (6, -15) \).
• Additionally, \( u \) is multiplied by 4: \( 4u = 4(-4, 3) = (-16, 12) \).

Notice how each component is scaled independently, allowing us to effectively increase or decrease the size of the vector without changing its direction (unless the scalar is negative, which would reverse the direction). This operation is used extensively in scaling transformations in physics and computer graphics.

Vector Addition

Vector addition is the process of adding two or more vectors together. It follows the rule of component-wise addition, where corresponding components of the vectors are added together.

Let's break it down:

• If \( a = (a_1, a_2) \) and \( b = (b_1, b_2) \), then their sum \( a + b \) will be a new vector \( (a_1 + b_1, a_2 + b_2) \).

In the exercise we are given, the results from scalar multiplication of the vector \( u \) were combined:

• First, add the vectors \( 2u = (-8, 6) \) and \( 4u = (-16, 12) \).
• Carry out the addition component-wise: \((-8 + (-16), 6 + 12) = (-24, 18) \).

This way of adding vectors is like following a path that combines the effects of the two vectors, moving first in the direction of one and then the other. It's also often visualized by placing the tail of one vector at the head of another and finding the resultant vector, which points from the tail of the first to the head of the last.

Vector Subtraction

Vector subtraction is similar to vector addition but instead involves subtracting the components of one vector from another. It's a straightforward extension of addition, providing a way to reverse a vector's direction before combining.

Here's how it's done:

• For vectors \( a = (a_1, a_2) \) and \( b = (b_1, b_2) \), the difference \( a - b \) is \( (a_1 - b_1, a_2 - b_2) \).

Continuing with the exercise, after performing the addition of vectors, the last step involved vector subtraction:

• From the vector obtained through addition \( (-24, 18) \), subtract \( 3v = (6, -15) \).
• Subtract these component-wise: \((-24 - 6, 18 - (-15)) = (-30, 33) \).

By viewing vector subtraction as adding the negative of a vector, we can understand it as following a path in the opposite direction, making it useful for finding relative positions, distances between points, or reversing movements in physics computations.