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Magazine Chapter 12: Problem 44
Find the scalar and vector projections of \(\mathbf{b}\) onto \(\mathbf{a}\) $$ a=i+2 j+3 k, \quad b=5 i-k $$
Short Answer
Expert verified
Scalar projection is \( \frac{2}{\sqrt{14}} \); vector projection is \( \frac{1}{7}i + \frac{2}{7}j + \frac{3}{7}k \).
Step by step solution
01
Understand the Concept of Projection
To find the projection of vector \( \mathbf{b} \) onto vector \( \mathbf{a} \), we need to determine two things: the scalar projection (also known as the component of \( \mathbf{b} \) along \( \mathbf{a} \)) and the vector projection (which is a vector along \( \mathbf{a} \) direction with the same magnitude).
02
Calculate the Dot Product of \( \mathbf{a} \) and \( \mathbf{b} \)
The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as:\[ \mathbf{a} \cdot \mathbf{b} = (1)(5) + (2)(0) + (3)(-1) = 5 + 0 - 3 = 2 \]
03
Find the Magnitude of \( \mathbf{a} \)
The magnitude of vector \( \mathbf{a} \) is:\[ \| \mathbf{a} \| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]
04
Compute the Scalar Projection of \( \mathbf{b} \) onto \( \mathbf{a} \)
Using the formula for scalar projection, which is given by:\[ \text{scalar projection} = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \|} = \frac{2}{\sqrt{14}} \]
05
Compute the Vector Projection of \( \mathbf{b} \) onto \( \mathbf{a} \)
The vector projection formula is:\[ \text{vector projection} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \|^2} \right) \mathbf{a} \]Substituting the values:\[ \text{vector projection} = \left( \frac{2}{14} \right) (i + 2j + 3k) = \frac{1}{7} (i + 2j + 3k) = \frac{1}{7}i + \frac{2}{7}j + \frac{3}{7}k \]
06
Review the Results of the Projections
Make sure to verify the calculations:- Scalar projection of \( \mathbf{b} \) onto \( \mathbf{a} \) is \( \frac{2}{\sqrt{14}} \).- Vector projection of \( \mathbf{b} \) onto \( \mathbf{a} \) is \( \frac{1}{7}i + \frac{2}{7}j + \frac{3}{7}k \).
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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, also known as scalar product, is a way to multiply two vectors to get a scalar value. Consider two vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \). The dot product is calculated as:\[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\]For example, the dot product of \( \mathbf{a} = i + 2j + 3k \) and \( \mathbf{b} = 5i - k \) equals:
- Multiply the respective components: \( (1)(5) + (2)(0) + (3)(-1) \)
- Add the products: \( 5 + 0 - 3 = 2 \)
Vector Magnitude
Vector magnitude is a measure of a vector's length or size. For any vector \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \), its magnitude is computed using the formula:\[\| \mathbf{a} \| = \sqrt{a_1^2 + a_2^2 + a_3^2}\]Considering our vector \( \mathbf{a} = i + 2j + 3k \):
- Square each component: \( 1^2, 2^2, 3^2 \)
- Add them together: \( 1 + 4 + 9 = 14 \)
- Take the square root: \( \sqrt{14} \)
Scalar Projection
Scalar projection is the 'shadow' or component of one vector along another vector's direction. It provides a measure of the strength or amount of one vector in the direction of another vector.The scalar projection of vector \( \mathbf{b} \) onto vector \( \mathbf{a} \) is calculated using:\[\text{scalar projection} = \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \|}\]For our example with \( \mathbf{a} \cdot \mathbf{b} = 2 \) and \( \|\mathbf{a}\| = \sqrt{14} \):
- Divide the dot product by the magnitude: \( \frac{2}{\sqrt{14}} \)
Projection Formulas
Projection formulas are key in determining both the scalar and vector projections of one vector on another. The vector projection specifically gives us a vector that lies in the direction of \( \mathbf{a} \) with a length equal to the scalar projection.The vector projection formula is:\[\text{vector projection of } \mathbf{b} \text{ onto } \mathbf{a} = \left( \frac{\mathbf{a} \cdot \mathbf{b}}{\| \mathbf{a} \|^2} \right) \mathbf{a}\]For our vectors, this comes down to:
- Calculate \( \frac{2}{14} = \frac{1}{7} \)
- Multiply with each component of \( \mathbf{a} = i + 2j + 3k \)
- Resulting in: \( \frac{1}{7}i + \frac{2}{7}j + \frac{3}{7}k \)
