Calculating 2w - Z With Vectors A Step-by-Step Solution

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In the realm of mathematics, particularly in linear algebra, vector operations play a crucial role. Understanding how to manipulate vectors is fundamental to various applications, ranging from physics simulations to computer graphics. This article delves into the process of vector subtraction and scalar multiplication, providing a step-by-step guide to solving the problem: If w=⟨3.5,−6⟩w = \langle 3.5, -6 \rangle and z=⟨−1.5,−4⟩z = \langle -1.5, -4 \rangle, what is the resulting vector for 2w−z2w - z?

Understanding Vectors and Vector Operations

Before diving into the solution, it's essential to grasp the basic concepts of vectors and vector operations. A vector is a mathematical object that has both magnitude and direction. It is often represented as an ordered list of numbers, called components. In two-dimensional space, a vector can be written as ⟨x,y⟩\langle x, y \rangle, where x and y are the horizontal and vertical components, respectively. Vector operations involve manipulating these vectors using arithmetic operations such as addition, subtraction, and scalar multiplication.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (a real number). This operation scales the magnitude of the vector without changing its direction (unless the scalar is negative, in which case the direction is reversed). If v=⟨x,y⟩v = \langle x, y \rangle and c is a scalar, then the scalar product cv is given by:

cv=c⟨x,y⟩=⟨cx,cy⟩cv = c \langle x, y \rangle = \langle cx, cy \rangle

In other words, each component of the vector is multiplied by the scalar.

Vector Subtraction

Vector subtraction involves subtracting one vector from another. If u=⟨x1,y1⟩u = \langle x_1, y_1 \rangle and v=⟨x2,y2⟩v = \langle x_2, y_2 \rangle, then the difference u - v is given by:

u−v=⟨x1−x2,y1−y2⟩u - v = \langle x_1 - x_2, y_1 - y_2 \rangle

Each corresponding component of the vectors is subtracted.

Step-by-Step Solution for 2w - z

Now, let's apply these concepts to solve the problem at hand. We are given two vectors: w=⟨3.5,−6⟩w = \langle 3.5, -6 \rangle and z=⟨−1.5,−4⟩z = \langle -1.5, -4 \rangle. Our goal is to find the resulting vector for 2w−z2w - z. This involves two operations: scalar multiplication (multiplying w by 2) and vector subtraction (subtracting z from 2w).

Step 1: Scalar Multiplication (2w)

First, we multiply vector w by the scalar 2:

2w=2⟨3.5,−6⟩=⟨2×3.5,2×−6⟩=⟨7,−12⟩2w = 2 \langle 3.5, -6 \rangle = \langle 2 \times 3.5, 2 \times -6 \rangle = \langle 7, -12 \rangle

So, 2w=⟨7,−12⟩2w = \langle 7, -12 \rangle.

Step 2: Vector Subtraction (2w - z)

Next, we subtract vector z from the result obtained in step 1:

2w−z=⟨7,−12⟩−⟨−1.5,−4⟩=⟨7−(−1.5),−12−(−4)⟩2w - z = \langle 7, -12 \rangle - \langle -1.5, -4 \rangle = \langle 7 - (-1.5), -12 - (-4) \rangle

Now, perform the subtractions:

⟨7−(−1.5),−12−(−4)⟩=⟨7+1.5,−12+4⟩=⟨8.5,−8⟩\langle 7 - (-1.5), -12 - (-4) \rangle = \langle 7 + 1.5, -12 + 4 \rangle = \langle 8.5, -8 \rangle

Therefore, the resulting vector for 2w−z2w - z is ⟨8.5,−8⟩\langle 8.5, -8 \rangle.

Analyzing the Options

Now, let's compare our result with the given options:

  • A. ⟨5,−2⟩\langle 5, -2 \rangle
  • B. ⟨10,−4⟩\langle 10, -4 \rangle
  • C. ⟨8.5,−8⟩\langle 8.5, -8 \rangle

Our calculated result, ⟨8.5,−8⟩\langle 8.5, -8 \rangle, matches option C. Therefore, the correct answer is:

  • C. ⟨8.5,−8⟩\langle 8.5, -8 \rangle

Common Mistakes and How to Avoid Them

When performing vector operations, it's easy to make mistakes if you're not careful. Here are some common pitfalls and how to avoid them:

1. Incorrect Scalar Multiplication

Mistake: Forgetting to multiply all components of the vector by the scalar.

How to Avoid: Ensure that you multiply each component of the vector by the scalar. For example, when calculating 2w, make sure to multiply both the x-component and the y-component of w by 2.

2. Incorrect Vector Subtraction

Mistake: Subtracting the components in the wrong order or making sign errors.

How to Avoid: Remember that vector subtraction is component-wise. Always subtract the corresponding components in the correct order. Pay close attention to signs, especially when subtracting negative numbers.

3. Misunderstanding the Order of Operations

Mistake: Performing vector subtraction before scalar multiplication.

How to Avoid: Follow the correct order of operations (PEMDAS/BODMAS). In this case, perform scalar multiplication before vector subtraction. For example, in the expression 2w - z, calculate 2w first, and then subtract z.

4. Not Double-Checking the Calculations

Mistake: Making arithmetic errors and not verifying the result.

How to Avoid: After performing the calculations, take a moment to double-check your work. Ensure that you haven't made any arithmetic errors, such as incorrect addition or subtraction.

Importance of Vector Operations in Real-World Applications

Vector operations are not just theoretical concepts; they have numerous practical applications in various fields. Here are a few examples:

1. Physics

In physics, vectors are used to represent physical quantities such as force, velocity, and acceleration. Vector operations are essential for analyzing the motion of objects, calculating the net force acting on an object, and understanding the interactions between objects. For example, when analyzing projectile motion, vectors are used to represent the initial velocity, gravitational force, and air resistance. Vector addition and subtraction are used to determine the resultant velocity and position of the projectile at any given time.

2. Computer Graphics

In computer graphics, vectors are used to represent points, directions, and transformations in 3D space. Vector operations are fundamental to rendering images, creating animations, and simulating realistic lighting effects. For example, when rendering a 3D scene, vectors are used to define the vertices of objects, the direction of light sources, and the camera position. Vector transformations, such as rotation and scaling, are used to manipulate the objects in the scene. Dot products and cross products are used to calculate lighting and shading effects.

3. Engineering

In engineering, vectors are used to analyze forces, stresses, and strains in structures and mechanical systems. Vector operations are crucial for designing bridges, buildings, and machines that can withstand various loads and stresses. For example, when designing a bridge, engineers use vectors to represent the forces acting on the bridge, such as the weight of the bridge itself, the weight of vehicles, and wind loads. Vector addition and subtraction are used to determine the resultant forces and moments acting on the bridge. Finite element analysis, a widely used engineering technique, relies heavily on vector operations to solve complex structural problems.

4. Robotics

In robotics, vectors are used to represent the position, orientation, and velocity of robots and their components. Vector operations are essential for controlling the movement of robots, planning paths, and coordinating multiple robots. For example, when programming a robot to pick up an object, vectors are used to define the position and orientation of the object, the robot's end-effector, and the desired trajectory. Vector transformations are used to convert between different coordinate systems and to calculate the joint angles required to reach the desired position. Path planning algorithms often use vector operations to find the shortest or most efficient path for a robot to move between two points.

Conclusion

In conclusion, understanding vector operations is crucial for success in mathematics and various STEM fields. By mastering concepts such as scalar multiplication and vector subtraction, you can solve a wide range of problems and gain a deeper appreciation for the power of vectors. In the given problem, we successfully calculated 2w−z2w - z by first performing scalar multiplication and then vector subtraction, arriving at the correct answer: ⟨8.5,−8⟩\langle 8.5, -8 \rangle. By avoiding common mistakes and practicing regularly, you can confidently tackle vector operations in any context. The applications of vector operations extend far beyond the classroom, making them an essential tool for scientists, engineers, and anyone working with quantitative data.

This detailed explanation not only provides the correct answer but also equips you with a comprehensive understanding of the underlying concepts and their practical applications. Keep practicing, and you'll become proficient in vector operations in no time!