Question

Find the vector \(\vec{C}\) that satisfies the equation \(3 \hat{x}+6 \hat{y}\) \(10 \hat{z}+\vec{C}=-7 \hat{x}+14 \hat{y}\)

Short answer

Answer: The vector \(\vec{C}\) that satisfies the given equation is \(\vec{C} = -10 \hat{x} + 8 \hat{y} - 10 \hat{z}\).

Step-by-step solution

Write the given equation with vectors in component form

The equation provided is: \(3 \hat{x} + 6 \hat{y} + 10 \hat{z} + \vec{C} = -7 \hat{x} + 14 \hat{y}\) We can rewrite this equation in component form as follows: \((3, 6, 10) + \vec{C} = (-7, 14, 0)\)

Isolate vector \(\vec{C}\)

In order to determine vector \(\vec{C}\), we need to isolate it in the equation. We will subtract the vector \((3, 6, 10)\) from both sides of the equation: \(\vec{C} = (-7, 14, 0) - (3, 6, 10)\)

Subtract the vectors

Now, subtract the corresponding components in the two vectors on the right-hand side of the equation: \(\vec{C} = (-7-3, 14-6, 0-10)\)

Simplify the result

Finally, simplify the numerical operations in the components: \(\vec{C} = (-10, 8, -10)\) So, the vector \(\vec{C}\) that satisfies the given equation is \(\vec{C} = -10 \hat{x} + 8 \hat{y} - 10 \hat{z}\).

Key concepts

Vector Addition

Vector addition is like combining forces. Imagine two arrows pointing in space, each representing a vector. By adding them, you’re essentially finding a new arrow that combines their lengths and directions. You line up these vectors, tail to tip, and the arrow from the beginning of the first to the tip of the second is your resultant vector.

To add vectors in component form, simply add their corresponding components:

• The x-components are added together
• The y-components are added together
• The z-components are added together if the vectors are 3D

For example, if you have \(\vec{A} = (a_x, a_y, a_z)\) and \(\vec{B} = (b_x, b_y, b_z)\), the sum \(\vec{A} + \vec{B}\) would be \( (a_x + b_x, a_y + b_y, a_z + b_z)\). This is how you combine their efforts into one unified direction.

Vector Subtraction

Vector subtraction works similarly to addition; however, it’s about taking the difference between two vectors. Subtracting one vector from another can be visualized as finding the vector that, when added to the second vector, yields the first vector.

In component form, you perform subtraction component by component:

• Subtract the x-components
• Subtract the y-components
• Subtract the z-components, if applicable

Let’s say you have vectors \(\vec{A} = (a_x, a_y, a_z)\) and \(\vec{B} = (b_x, b_y, b_z)\). The result of \(\vec{A} - \vec{B}\) will be \( (a_x - b_x, a_y - b_y, a_z - b_z)\). This operation is essential for isolating a vector in equations, as shown in our initial exercise.

Component Form

The component form of a vector expresses it in terms of its individual parts, or components, along each axis in a coordinate system.

This way of representing vectors breaks them down into \(x\), \(y\), and \(z\) components: \( (x, y, z)\) for 3D vectors, or just \( (x, y)\) for 2D vectors. By separating vectors into components, mathematical operations like addition and subtraction become straightforward: you simply handle each component individually.

Think of it as splitting the vector into its horizontal and vertical effects—making complex multi-dimensional movements manageable. For example, the vector \( (3, 6, 10)\) implies it moves 3 units along the x-axis, 6 along the y-axis, and 10 along the z-axis.

Three-Dimensional Vectors

Three-dimensional vectors describe quantities that have magnitude and direction in a 3D space. This is like adding a new layer to the usual 2D plane, letting you represent directions and movements in real-world space more accurately. Each vector in three dimensions can be broken down into three components: \( (x, y, z) \).

These components tell you how far a vector travels along each axis:

• \(x\)-axis: horizontal movement
• \(y\)-axis: vertical movement
• \(z\)-axis: depth movement (in and out of the page)

Understanding vectors in 3D is crucial when working with physics problems or in engineering designs, because it allows for a comprehensive description of an object’s position or a force's impact in space. For example, solving \(\vec{C} = (-10, 8, -10)\), involves working in 3D as each component determines distinct spatial directions.