Problem 20
Question
Determine whether the given vectors are perpendicular. $$\mathbf{u}=4 \mathbf{i}, \quad \mathbf{v}=-\mathbf{i}+3 \mathbf{j}$$
Step-by-Step Solution
Verified Answer
The vectors are not perpendicular.
1Step 1: Determine the components of the vectors
The vector \( \mathbf{u} = 4 \mathbf{i} \) only has an \( \mathbf{i} \) component, which means its components are \( (4, 0) \). The vector \( \mathbf{v} = -\mathbf{i} + 3 \mathbf{j} \) has components \( (-1, 3) \).
2Step 2: Use the dot product formula
The dot product of two vectors \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \) is given by the formula \( a_1b_1 + a_2b_2 \). For vectors to be perpendicular, their dot product must equal zero.
3Step 3: Calculate the dot product
Substitute the components of \( \mathbf{u} \) and \( \mathbf{v} \) into the formula: \( 4(-1) + 0(3) = -4 + 0 = -4 \).
4Step 4: Analyze the result
Since the dot product is \(-4\) and not zero, vectors \( \mathbf{u} \) and \( \mathbf{v} \) are not perpendicular.
Key Concepts
Components of VectorsDot ProductVectors in Mathematics
Components of Vectors
To understand vectors, it's essential to break them down into their components. A vector in mathematics is often represented in a base coordinate system using unit vectors, commonly denoted by \( \mathbf{i} \) and \( \mathbf{j} \) in two-dimensional space. The vector's components are essentially the projections along these axes.
In the given exercise, vector \( \mathbf{u} = 4 \mathbf{i} \) has components \( (4, 0) \), meaning it moves 4 units in the x-direction and 0 units in the y-direction. Meanwhile, vector \( \mathbf{v} = -\mathbf{i} + 3 \mathbf{j} \) has components \( (-1, 3) \), indicating a movement of -1 unit in the x-direction and 3 units in the y-direction.
Breaking vectors into components is crucial for calculations in physics and engineering. It helps us analyze vector relationships, like determining collinearity or orthogonality.
In the given exercise, vector \( \mathbf{u} = 4 \mathbf{i} \) has components \( (4, 0) \), meaning it moves 4 units in the x-direction and 0 units in the y-direction. Meanwhile, vector \( \mathbf{v} = -\mathbf{i} + 3 \mathbf{j} \) has components \( (-1, 3) \), indicating a movement of -1 unit in the x-direction and 3 units in the y-direction.
Breaking vectors into components is crucial for calculations in physics and engineering. It helps us analyze vector relationships, like determining collinearity or orthogonality.
Dot Product
The dot product, also known as the scalar product, is an essential concept when dealing with vectors. It is used to determine the degree of perpendicularity between two vectors. When the dot product of two vectors is zero, the vectors are considered perpendicular.
The dot product of two vectors, say \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), is calculated using the formula:
In our exercise, applying the dot product formula to vectors \( \mathbf{u} \) and \( \mathbf{v} \), we calculated \(-4\), confirming these vectors are not perpendicular since the dot product isn't zero.
The dot product of two vectors, say \( \mathbf{a} = (a_1, a_2) \) and \( \mathbf{b} = (b_1, b_2) \), is calculated using the formula:
- \( a_1b_1 + a_2b_2 \)
In our exercise, applying the dot product formula to vectors \( \mathbf{u} \) and \( \mathbf{v} \), we calculated \(-4\), confirming these vectors are not perpendicular since the dot product isn't zero.
Vectors in Mathematics
Vectors are fundamental elements in mathematics and physics, representing quantities that have both magnitude and direction. Unlike scalars, which only have magnitude, vectors allow for precise modeling of various phenomena, such as force, velocity, and acceleration.
In mathematics, vectors are often used to perform operations such as addition, subtraction, and multiplication. One common method of vector multiplication is the dot product, which helps determine angles and perpendicularity. Another is the cross product, used primarily in three-dimensional space to find a vector perpendicular to two given vectors.
Understanding vectors and their operations is crucial in various fields, including engineering, computer graphics, and navigation. Vectors provide a way to map out forces acting on an object, simulate movement, or rotate objects in a virtual environment. Once you master vector mathematics, it'll be easier to handle real-world applications efficiently.
In mathematics, vectors are often used to perform operations such as addition, subtraction, and multiplication. One common method of vector multiplication is the dot product, which helps determine angles and perpendicularity. Another is the cross product, used primarily in three-dimensional space to find a vector perpendicular to two given vectors.
Understanding vectors and their operations is crucial in various fields, including engineering, computer graphics, and navigation. Vectors provide a way to map out forces acting on an object, simulate movement, or rotate objects in a virtual environment. Once you master vector mathematics, it'll be easier to handle real-world applications efficiently.
Other exercises in this chapter
Problem 20
Express the given vector in terms of the unit vectors i, \(\mathbf{j}\). and \(\mathbf{k}\). $$\langle 0,-3,5\rangle$$
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Describe the trace of the sphere $$x^{2}+(y-4)^{2}+(z-3)^{2}=144$$ in (a) the \(x z\) -plane and in (b) the plane \(z=-2\)
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Sketch the given vector with initial point (4, 3), and find the terminal point. $$\mathbf{u}=\langle- 1,2\rangle$$
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Find the area of the parallelogram determined by the given vectors. $$\mathbf{u}=\langle 3,2,1\rangle, \quad \mathbf{v}=\langle 1,2,3\rangle$$
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