Problem 3
Question
Sketch vector in an \(x\)-\(y\) coordinate system, and compute the length of the vector. $$\langle-4,2\rangle$$
Step-by-Step Solution
Verified Answer
The vector \(\langle -4, 2 \rangle\) has length \(2\sqrt{5}\).
1Step 1: Understand the Vector Components
The vector given is \( \langle -4, 2 \rangle \). It consists of two components: \(-4\) in the \(x\) direction and \(2\) in the \(y\) direction.
2Step 2: Sketch the Vector
On an \(x\)-\(y\) coordinate plane, start from the origin \((0,0)\). Move \(4\) units to the left (as \(x = -4\)) and \(2\) units up (as \(y = 2\)) to reach the point \((-4, 2)\). Draw an arrow from the origin to \((-4, 2)\) to represent the vector.
3Step 3: Find the Length of the Vector
The length (magnitude) of a vector \( \langle a, b \rangle \) is calculated using the formula: \( \sqrt{a^2 + b^2} \). For \( \langle -4, 2 \rangle \), substitute \(a = -4\) and \(b = 2\): \[\text{Length} = \sqrt{(-4)^2 + 2^2} = \sqrt{16 + 4} = \sqrt{20}\]Simplify \(\sqrt{20}\):\[\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2 \sqrt{5}\]
4Step 4: Conclusion
The length of the vector \( \langle -4, 2 \rangle \) is \(2\sqrt{5}\) and it points towards \((-4, 2)\) in the coordinate system.
Key Concepts
Understanding Coordinate SystemsVector Components ExplainedSketching VectorsCalculating Vector Magnitude
Understanding Coordinate Systems
To work with vectors, the first step is to understand the coordinate system. Coordinate systems allow us to precisely describe the position of points or vectors in space. The most common coordinate system is the Cartesian coordinate system, which uses two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point on this plane is represented by an ordered pair \(x, y\), indicating its position relative to the origin \(0, 0\).
By utilizing this system, we can translate real-world problems into mathematical expressions. This transformation makes it easier to perform calculations like vector addition, subtraction, and finding magnitudes. In exercises like sketching vector \langle -4, 2 \rangle, we rely on the coordinate system to locate and draw the vector accurately.
By utilizing this system, we can translate real-world problems into mathematical expressions. This transformation makes it easier to perform calculations like vector addition, subtraction, and finding magnitudes. In exercises like sketching vector \langle -4, 2 \rangle, we rely on the coordinate system to locate and draw the vector accurately.
Vector Components Explained
Every vector is defined by its components, which represent its direction and magnitude within a coordinate system. For the vector \langle -4, 2 \rangle, these components are \(-4\) on the x-axis and \(2\) on the y-axis.
\- The x-component \(-4\) means that the vector moves 4 units to the left from the origin, indicating a negative x-direction.
\- The y-component \ (+2) means that the vector moves 2 units up, indicating a positive y-direction.
Understanding these components is crucial for vector operations. They help in determining the vector's position and length and are foundational for further calculations such as finding the magnitude or adding/subtracting vectors.
\- The x-component \(-4\) means that the vector moves 4 units to the left from the origin, indicating a negative x-direction.
\- The y-component \ (+2) means that the vector moves 2 units up, indicating a positive y-direction.
Understanding these components is crucial for vector operations. They help in determining the vector's position and length and are foundational for further calculations such as finding the magnitude or adding/subtracting vectors.
Sketching Vectors
Sketching a vector involves visually representing its direction and length on the coordinate plane. To sketch the vector \langle -4, 2 \rangle, we begin at the origin point \(0, 0\).
Follow these steps to draw it:
\- Move horizontally to the left by 4 units due to the x-component \(-4\).
\- Then move vertically upwards by 2 units because of the y-component \(2\).
\- Mark the point \(-4, 2\) on the plane, and draw an arrow from the origin to this point. This arrow represents the vector.
The sketch not only illustrates the vector's path but also gives an intuitive understanding of its position relative to other vectors. Visual aids like sketches are essential for comprehending vector geometry and solving complex problems with ease.
Follow these steps to draw it:
\- Move horizontally to the left by 4 units due to the x-component \(-4\).
\- Then move vertically upwards by 2 units because of the y-component \(2\).
\- Mark the point \(-4, 2\) on the plane, and draw an arrow from the origin to this point. This arrow represents the vector.
The sketch not only illustrates the vector's path but also gives an intuitive understanding of its position relative to other vectors. Visual aids like sketches are essential for comprehending vector geometry and solving complex problems with ease.
Calculating Vector Magnitude
The magnitude of a vector indicates its length, which is a measure of how far a point is from the origin in vector space. For a vector \langle a, b \rangle, its magnitude is denoted by \||\vec{v}||\, the symbol for the vector's length.
The formula for the magnitude of a vector is: \[ ||\vec{v}|| = \sqrt{a^2 + b^2} \]
Using the given vector \langle -4, 2 \rangle:
\- Compute \( (-4)^2 = 16 \) and \( 2^2 = 4 \).
\- Sum these squared components: \( 16 + 4 = 20 \).
\- Take the square root to find the magnitude: \( \sqrt{20} = 2\sqrt{5} \).
The calculated magnitude \(2\sqrt{5} \) gives us a measure of the vector's length. It's a crucial step in understanding how long the vector is, regardless of what direction it points.
The formula for the magnitude of a vector is: \[ ||\vec{v}|| = \sqrt{a^2 + b^2} \]
Using the given vector \langle -4, 2 \rangle:
\- Compute \( (-4)^2 = 16 \) and \( 2^2 = 4 \).
\- Sum these squared components: \( 16 + 4 = 20 \).
\- Take the square root to find the magnitude: \( \sqrt{20} = 2\sqrt{5} \).
The calculated magnitude \(2\sqrt{5} \) gives us a measure of the vector's length. It's a crucial step in understanding how long the vector is, regardless of what direction it points.
Other exercises in this chapter
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