Problem 21 - Precalculus

Question

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

Step-by-Step Solution

Answer

The resulting vector is \((0, -14)\).

Step 1: Understand the Scalar Multiplication

We need to multiply each component of vector \(u=(-4,3)\) by the scalar 2. This involves taking 2 times each of the components separately. Thus, we calculate \(2 \times (-4)\) and \(2 \times 3\).

Step 2: Perform Scalar Multiplication on vector u

Calculate the result of scaling vector \(u=(-4,3)\):\[2 \times (-4,3) = (2 \times -4, 2 \times 3) = (-8, 6)\]

Step 3: Understand the Scalar Multiplication for vector v

We now multiply vector \(v=\langle 2,-5\rangle\) by the scalar 4. This involves multiplying 4 with each component of \(v\), namely \(4 \times 2\) and \(4 \times -5\).

Step 4: Perform Scalar Multiplication on vector v

Calculate the result of scaling vector \(v=\langle 2,-5\rangle\):\[4 \times \langle 2,-5 \rangle = (4 \times 2, 4 \times -5) = (8, -20)\]

Step 5: Add the Resulting Vectors

Now, add the resultant vectors from the previous steps: \((-8, 6)\) and \((8, -20)\). To add them, simply add corresponding components: \[(-8 + 8, 6 + (-20)) = (0, -14)\]

Step 6: Conclusion

The result of the operation \(2u + 4v\) is the vector \((0, -14)\).

Key Concepts

Scalar Multiplication

Scalar multiplication in vector operations is a straightforward but essential concept. It involves multiplying each component of a vector by a scalar, which is essentially a single number. This process stretches or compresses the vector depending on the magnitude of the scalar.

For example, consider a vector \( u = (-4, 3) \). If we multiply it by a scalar 2, each component of the vector is multiplied by 2:

• The first component: \( 2 \times (-4) = -8 \)
• The second component: \( 2 \times 3 = 6 \)

This changes the vector to \( (-8, 6) \). Similarly, to perform scalar multiplication on vector \( v = \langle 2, -5 \rangle \) by a scalar 4, we multiply each component accordingly:

• The first component: \( 4 \times 2 = 8 \)
• The second component: \( 4 \times (-5) = -20 \)

This gives us a new vector \( (8, -20) \). Scalar multiplication is a building block for many vector operations and helps in scaling vectors.

Vector Addition

Vector addition involves combining two vectors to produce a third vector. This is done component-wise, meaning you add the corresponding components from each vector together. It’s like aligning two arrows and figuring out the direction of movement if you followed the first arrow then the second.

In our exercise, after performing scalar multiplication, we have two new vectors: \( (-8, 6) \) and \( (8, -20) \). To add these vectors, add their components:

• First components: \( -8 + 8 = 0 \)
• Second components: \( 6 + (-20) = -14 \)

So, the resulting vector from adding these two vectors is \( (0, -14) \). This method of adding vectors is crucial in physics and engineering for determining results like displacement or force resultant.

Vectors in Component Form

Understanding vectors in component form is foundational for performing vector operations. A vector in component form breaks down the vector into its parts along the x, y, (and possibly z) axes. It essentially describes how much the vector moves in each direction.

For example, a vector \( u = (-4, 3) \) in component form indicates it moves 4 units left (negative x-direction) and 3 units up (positive y-direction). Similarly, vector \( v = \langle 2, -5 \rangle \) moves 2 units right and 5 units down.
Vectors are written in bold in textbooks or as ordered pairs to denote this directional breakdown. This form allows for easy arithmetic operations like addition, subtraction, and scalar multiplication since it splits the vector into manageable components.

Understanding and using vectors in component form is especially important in fields that deal with directions and magnitudes, such as physics or computer science for graphical applications.