Problem 33
Question
Show that each pair of vectors is perpendicular. \(2 \mathbf{i}+\mathbf{j}\) and \(\mathbf{i}-2 \mathbf{j}\)
Step-by-Step Solution
Verified Answer
The vectors are perpendicular as their dot product is zero.
1Step 1: Understand the Definition
Two vectors are perpendicular if their dot product is zero. For vectors \(\mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \(\mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), the dot product is calculated as \(a_1 b_1 + a_2 b_2\).
2Step 2: Identify Components
The first vector is \(\mathbf{a} = 2\mathbf{i} + \mathbf{j}\) with components \(a_1 = 2\) and \(a_2 = 1\). The second vector is \(\mathbf{b} = \mathbf{i} - 2\mathbf{j}\) with components \(b_1 = 1\) and \(b_2 = -2\).
3Step 3: Calculate the Dot Product
Substitute these component values into the dot product formula: \(a_1 b_1 + a_2 b_2 = 2 \cdot 1 + 1 \cdot (-2)\).
4Step 4: Simplify the Dot Product
Calculate the result of the expression: \(2 \cdot 1 + 1 \cdot (-2) = 2 - 2 = 0\).
5Step 5: Conclude Perpendicularity
Since the dot product is zero, the vectors \(2 \mathbf{i} + \mathbf{j}\) and \(\mathbf{i} - 2 \mathbf{j}\) are perpendicular.
Key Concepts
Dot ProductVector ComponentsVector Mathematics
Dot Product
The dot product is a fundamental concept in vector mathematics. It is an operation that takes two equal-dimension vectors and produces a single scalar. Thinking of it as a way to multiply vectors, its result provides valuable insight into the relationship between the vectors.
Dot product is often used to determine if two vectors are perpendicular. Two vectors are perpendicular when their dot product equals zero. This is because the cosine of the angle between the vectors is zero, which corresponds to an angle of 90 degrees or \(\frac{\pi}{2}\) radians.
To calculate the dot product of vectors \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \), use the formula:
Dot product is often used to determine if two vectors are perpendicular. Two vectors are perpendicular when their dot product equals zero. This is because the cosine of the angle between the vectors is zero, which corresponds to an angle of 90 degrees or \(\frac{\pi}{2}\) radians.
To calculate the dot product of vectors \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \), use the formula:
- \(a_1 b_1 + a_2 b_2 \)
Vector Components
Vectors are like arrows. They have both a direction and a magnitude, which can be broken down into parts called components. These components correspond to the directions of an axis in a coordinate system, like the x-axis and y-axis in Cartesian coordinates.
A vector \(\mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} \) possesses two components: \(v_1\) which is aligned with the x-axis or \(\mathbf{i} \) unit vector, and \(v_2\), which is along the y-axis or \(\mathbf{j} \) unit vector. Each component is simply a multiplier that tells how far to go in each direction from the origin.
By identifying these components, you can reassemble a vector from them, and these values are essential when performing operations such as calculating the dot product. The components determine how each vector contributes to the overall operation or geometric interpretation.
A vector \(\mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} \) possesses two components: \(v_1\) which is aligned with the x-axis or \(\mathbf{i} \) unit vector, and \(v_2\), which is along the y-axis or \(\mathbf{j} \) unit vector. Each component is simply a multiplier that tells how far to go in each direction from the origin.
By identifying these components, you can reassemble a vector from them, and these values are essential when performing operations such as calculating the dot product. The components determine how each vector contributes to the overall operation or geometric interpretation.
Vector Mathematics
Vector mathematics is a critical part of understanding geometrical and physical concepts. It allows you to manipulate quantities that have both a direction and magnitude, which are not captured by simple scalars.
Mathematical operations with vectors include addition, subtraction, scalar multiplication, and finding products like the dot product. These operations help to solve a variety of geometric problems, such as finding whether two vectors are perpendicular or calculating angles between them.
The elegance of vector mathematics is in its abstraction and versatility. Especially in physics and engineering, vectors provide a way to represent complex quantities such as forces or velocities in a very manageable way.
Mathematical operations with vectors include addition, subtraction, scalar multiplication, and finding products like the dot product. These operations help to solve a variety of geometric problems, such as finding whether two vectors are perpendicular or calculating angles between them.
The elegance of vector mathematics is in its abstraction and versatility. Especially in physics and engineering, vectors provide a way to represent complex quantities such as forces or velocities in a very manageable way.
- Vector Addition: Combine vectors by adding their respective components.
- Scalar Multiplication: Multiply each component by a scalar value to alter the vector's magnitude.
- Dot Product: A combination that results in a scalar, revealing directions and potential perpendicularity.
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