Chapter 1: Problem 15
For the two vectors \(A=(2,1,0)\) and \(B=(0,1,2),\) what is their scalar product, \(\vec{A} \cdot \vec{B} ?\) a) b) 6 c) 2 d) ( e)
Short Answer
Expert verified
Answer: 1
Step by step solution
01
Identify the components of the given vectors
The given vectors A and B are defined as follows:
A = (2, 1, 0)
B = (0, 1, 2)
02
Calculate the scalar product of the two vectors
Use the definition of the scalar product for 3-dimensional vectors:
\(\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z\)
Substitute the components of the given vectors A and B:
\(\vec{A} \cdot \vec{B} = (2)(0) + (1)(1) + (0)(2)\)
03
Simplify the calculated scalar product
Simplify the scalar product equation:
\(\vec{A} \cdot \vec{B} = 0 + 1 + 0 = 1\)
So, the scalar product of the two given vectors A and B is 1. However, this answer is not present in the given options. It might be a typo in the problem statement or the options provided. In any case, we have shown the step-by-step solution to find the scalar product of the given vectors.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Multiplication
The process of multiplying vectors can be performed in more than one way, and it is crucial to understand the type of multiplication involved. In vector multiplication, you encounter operations such as the dot product (also known as scalar product), and the cross product, which yields another vector.
When you multiply vectors, you take the components of each vector and combine them according to the rules of the operation. For instance, the dot product involves multiplying corresponding components of two vectors and summing the results. Let’s consider the vectors from the exercise, \(\vec{A} = (2,1,0)\) and \(\vec{B} = (0,1,2)\). In the dot product, we multiply 2 by 0, 1 by 1, and 0 by 2, then add up those products to get the scalar result.
When you multiply vectors, you take the components of each vector and combine them according to the rules of the operation. For instance, the dot product involves multiplying corresponding components of two vectors and summing the results. Let’s consider the vectors from the exercise, \(\vec{A} = (2,1,0)\) and \(\vec{B} = (0,1,2)\). In the dot product, we multiply 2 by 0, 1 by 1, and 0 by 2, then add up those products to get the scalar result.
Components of Vectors
Each vector in a Cartesian coordinate system is described by its components. These components are projections of the vector along the axes of the system. For 3-dimensional vectors, like the ones in our exercise, \(\vec{A} = (2,1,0)\) and \(\vec{B} = (0,1,2)\), there are three components corresponding to the x, y, and z axes.
In many physics and engineering problems, understanding these components is vital because they can represent different physical quantities, such as force, velocity, or displacement in each direction of three-dimensional space. By breaking these vectors into components, we simplify operations like addition, subtraction, and multiplication.
In many physics and engineering problems, understanding these components is vital because they can represent different physical quantities, such as force, velocity, or displacement in each direction of three-dimensional space. By breaking these vectors into components, we simplify operations like addition, subtraction, and multiplication.
3-Dimensional Vectors
A 3-dimensional vector has three distinct components which determine its magnitude and direction in three-dimensional space. These components correspond to the vector's projection onto the x, y, and z axes. The vector \(\vec{A} = (2,1,0)\) from the given problem has a component of 2 in the x-axis direction, 1 in the y-axis direction, and 0 in the z-axis direction, indicating that it lies entirely within the x-y plane.
Understanding 3-dimensional vectors is pivotal for visualizing geometrical concepts and solving problems in fields such as physics, engineering, and computer graphics. It’s important to comfortably navigate with these vectors, interpreting their components to understand their behavior in space.
Understanding 3-dimensional vectors is pivotal for visualizing geometrical concepts and solving problems in fields such as physics, engineering, and computer graphics. It’s important to comfortably navigate with these vectors, interpreting their components to understand their behavior in space.
Dot Product
The dot product, also known as scalar product, is a fundamental operation in vector algebra. It takes two vectors and returns a single scalar quantity. Unlike multiplication of regular numbers, the dot product encapsulates both the magnitude of the vectors and the cosine of the angle between them.
To calculate the dot product for vectors \(\vec{A}\) and \(\vec{B}\), you multiply their corresponding components and then sum these products: \(\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z\). In our example, when we applied this formula to vectors \(\vec{A} = (2,1,0)\) and \(\vec{B} = (0,1,2)\), we obtained the scalar result of 1.
This operation is widely used to find the angle between two vectors, project one vector onto another, or determine if two vectors are perpendicular (orthogonal), as the dot product will be zero if the vectors are at a 90-degree angle to each other.
To calculate the dot product for vectors \(\vec{A}\) and \(\vec{B}\), you multiply their corresponding components and then sum these products: \(\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y + A_zB_z\). In our example, when we applied this formula to vectors \(\vec{A} = (2,1,0)\) and \(\vec{B} = (0,1,2)\), we obtained the scalar result of 1.
This operation is widely used to find the angle between two vectors, project one vector onto another, or determine if two vectors are perpendicular (orthogonal), as the dot product will be zero if the vectors are at a 90-degree angle to each other.