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Magazine Chapter 8: Problem 18
Find the projection of \( \langle 0,4\rangle\)onto \(\langle 3,7\rangle .\)
Short Answer
Expert verified
The projection is \( \left( \frac{42}{29}, \frac{98}{29} \right) \)."
Step by step solution
01
Determine the Formula for Projection
The projection of a vector \( \mathbf{b} \) onto another vector \( \mathbf{a} \) is given by the formula: \[ \text{proj}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a} \] Here, \( \mathbf{b} = \langle 0,4 \rangle \) and \( \mathbf{a} = \langle 3,7 \rangle \).
02
Calculate the Dot Product of \(\mathbf{a}\) and \(\mathbf{b}\)
The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as follows:\[ \langle 3, 7 \rangle \cdot \langle 0, 4 \rangle = 3 \times 0 + 7 \times 4 = 28 \]
03
Calculate the Dot Product of \(\mathbf{a}\) with Itself
The dot product \( \mathbf{a} \cdot \mathbf{a} \) is:\[ \langle 3, 7 \rangle \cdot \langle 3, 7 \rangle = 3 \times 3 + 7 \times 7 = 9 + 49 = 58 \]
04
Compute the Projection Scalar
Use the results from the dot products to find the scalar that scales \( \mathbf{a} \):\[ \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{a} \cdot \mathbf{a}} = \frac{28}{58} = \frac{14}{29} \]
05
Determine the Projection Vector
Use the scalar to find the actual projection vector:\[ \left( \frac{14}{29} \right) \langle 3, 7 \rangle = \left( \frac{42}{29}, \frac{98}{29} \right) \] So, the projection of \( \langle 0, 4 \rangle \) onto \( \langle 3, 7 \rangle \) is \( \left( \frac{42}{29}, \frac{98}{29} \right) \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Dot Product
The dot product is a central operation when working with vectors. It is a way to multiply two vectors resulting in a scalar value. For two vectors, \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\), the dot product is calculated as follows:
- Multiply the corresponding components: \(u_1 \times v_1\) and \(u_2 \times v_2\).
- Add the results of these multiplications: \(u_1 \times v_1 + u_2 \times v_2\).
- First component: \(3 \times 0 = 0\)
- Second component: \(7 \times 4 = 28\)
Vector Operations
Vector operations are basic algebraic procedures applied to vectors, providing various ways to manipulate and understand them. Two common vector operations are addition and scalar multiplication:
- Addition: To add two vectors, sum their corresponding components. If \(\mathbf{u} = \langle u_1, u_2 \rangle\) and \(\mathbf{v} = \langle v_1, v_2 \rangle\), then \(\mathbf{u} + \mathbf{v} = \langle u_1 + v_1, u_2 + v_2 \rangle\).
- Scalar Multiplication: To multiply a vector by a scalar, multiply each component of the vector by that scalar. For a scalar \(k\) and vector \(\mathbf{u} = \langle u_1, u_2 \rangle\), \(k\mathbf{u} = \langle k \times u_1, k \times u_2 \rangle\).
Projection Formula
The projection formula allows you to find the shadow or projection of one vector onto another. For vectors \(\mathbf{b}\) and \(\mathbf{a}\), the projection of \(\mathbf{b}\) onto \(\mathbf{a}\) is given by:\[\text{proj}_{\mathbf{a}} \mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{a} \cdot \mathbf{a}} \mathbf{a}\]This can be broken down into several key steps:
- Calculate the dot product of \(\mathbf{a}\) and \(\mathbf{b}\), which gives you an idea of how much of \(\mathbf{b}\) aligns with \(\mathbf{a}\).
- Calculate the dot product of \(\mathbf{a}\) with itself, which is used to normalize \(\mathbf{a}\).
- Divide the result of the first dot product by the second, providing the scalar for projection.
- Multiply this scalar by \(\mathbf{a}\), resulting in the projection vector.
