7.3 Q-2

Question Statement

Calculate the projection of vector $\underline{\mathbf{a}}$ along $\underline{\mathbf{b}}$ and the projection of $\underline{\mathbf{b}}$ along $\underline{\mathbf{a}}$ for the following pairs of vectors:

(i) $\underline{\mathbf{a}} = \underline{\mathbf{i}} - \underline{\mathbf{k}}, \quad \underline{\mathbf{b}} = \underline{\mathbf{j}} + \underline{\mathbf{k}}$

(ii) $\underline{\mathbf{a}} = 3\underline{\mathbf{i}} + \underline{\mathbf{j}} - \underline{\mathbf{k}}, \quad \underline{\mathbf{b}} = -2\underline{\mathbf{i}} - \underline{\mathbf{j}} + \underline{\mathbf{k}}$

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Background and Explanation

The projection of one vector onto another is a measure of how much one vector “points” in the direction of another. The formula for the projection of $\underline{\mathbf{a}}$ along $\underline{\mathbf{b}}$ is:

$$\text{Projection of } \underline{\mathbf{a}} \text{ onto } \underline{\mathbf{b}} = \frac{\underline{\mathbf{a}} \cdot \underline{\mathbf{b}}}{|\underline{\mathbf{b}}|}$$

Similarly, the projection of $\underline{\mathbf{b}}$ along $\underline{\mathbf{a}}$ is:

$$\text{Projection of } \underline{\mathbf{b}} \text{ onto } \underline{\mathbf{a}} = \frac{\underline{\mathbf{a}} \cdot \underline{\mathbf{b}}}{|\underline{\mathbf{a}}|}$$

Where:

• $\underline{\mathbf{a}} \cdot \underline{\mathbf{b}}$ is the dot product of the two vectors.
• $|\underline{\mathbf{a}}|$ and $|\underline{\mathbf{b}}|$ are the magnitudes (lengths) of the vectors.

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Solution

(i) $\underline{\mathbf{a}} = \underline{\mathbf{i}} - \underline{\mathbf{k}}, \quad \underline{\mathbf{b}} = \underline{\mathbf{j}} + \underline{\mathbf{k}}$

1. Calculate the magnitudes of $\underline{\mathbf{a}}$ and $\underline{\mathbf{b}}$:

$$|\underline{\mathbf{a}}| = \sqrt{(1)^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2}$$ $$|\underline{\mathbf{b}}| = \sqrt{(1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2}$$

2. Calculate the dot product of $\underline{\mathbf{a}}$ and $\underline{\mathbf{b}}$:

$$\underline{\mathbf{a}} \cdot \underline{\mathbf{b}} = (1)(0) + (0)(1) + (-1)(1) = 0 + 0 - 1 = -1$$

3. Find the projections:
   • Projection of $\underline{\mathbf{a}}$ along $\underline{\mathbf{b}}$:

$$\text{Projection of } \underline{\mathbf{a}} \text{ onto } \underline{\mathbf{b}} = \frac{\underline{\mathbf{a}} \cdot \underline{\mathbf{b}}}{|\underline{\mathbf{b}}|} = \frac{-1}{\sqrt{2}}$$

• Projection of $\underline{\mathbf{b}}$ along $\underline{\mathbf{a}}$:

$$\text{Projection of } \underline{\mathbf{b}} \text{ onto } \underline{\mathbf{a}} = \frac{\underline{\mathbf{a}} \cdot \underline{\mathbf{b}}}{|\underline{\mathbf{a}}|} = \frac{-1}{\sqrt{2}}$$

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(ii) $\underline{\mathbf{a}} = 3\underline{\mathbf{i}} + \underline{\mathbf{j}} - \underline{\mathbf{k}}, \quad \underline{\mathbf{b}} = -2\underline{\mathbf{i}} - \underline{\mathbf{j}} + \underline{\mathbf{k}}$

1. Calculate the magnitudes of $\underline{\mathbf{a}}$ and $\underline{\mathbf{b}}$:

$$|\underline{\mathbf{a}}| = \sqrt{(3)^2 + (1)^2 + (-1)^2} = \sqrt{9 + 1 + 1} = \sqrt{11}$$ $$|\underline{\mathbf{b}}| = \sqrt{(-2)^2 + (-1)^2 + (1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6}$$

2. Calculate the dot product of $\underline{\mathbf{a}}$ and $\underline{\mathbf{b}}$:

$$\underline{\mathbf{a}} \cdot \underline{\mathbf{b}} = (3)(-2) + (1)(-1) + (-1)(1) = -6 - 1 - 1 = -8$$

3. Find the projections:
   • Projection of $\underline{\mathbf{a}}$ along $\underline{\mathbf{b}}$:

$$\text{Projection of } \underline{\mathbf{a}} \text{ onto } \underline{\mathbf{b}} = \frac{\underline{\mathbf{a}} \cdot \underline{\mathbf{b}}}{|\underline{\mathbf{b}}|} = \frac{-8}{\sqrt{6}}$$

• Projection of $\underline{\mathbf{b}}$ along $\underline{\mathbf{a}}$:

$$\text{Projection of } \underline{\mathbf{b}} \text{ onto } \underline{\mathbf{a}} = \frac{\underline{\mathbf{a}} \cdot \underline{\mathbf{b}}}{|\underline{\mathbf{a}}|} = \frac{-8}{\sqrt{11}}$$

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Key Formulas or Methods Used

• Projection of a vector onto another:

$$\text{Projection of } \underline{\mathbf{a}} \text{ onto } \underline{\mathbf{b}} = \frac{\underline{\mathbf{a}} \cdot \underline{\mathbf{b}}}{|\underline{\mathbf{b}}|}$$ $$\text{Projection of } \underline{\mathbf{b}} \text{ onto } \underline{\mathbf{a}} = \frac{\underline{\mathbf{a}} \cdot \underline{\mathbf{b}}}{|\underline{\mathbf{a}}|}$$

• Dot product:

$$\underline{\mathbf{a}} \cdot \underline{\mathbf{b}} = a_1b_1 + a_2b_2 + a_3b_3$$

• Magnitude of a vector:

$$|\underline{\mathbf{a}}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$$

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Summary of Steps

1. Calculate the magnitudes of both vectors $\underline{\mathbf{a}}$ and $\underline{\mathbf{b}}$.
2. Compute the dot product of the vectors.
3. Find the projection of $\underline{\mathbf{a}}$ along $\underline{\mathbf{b}}$ using the formula $\frac{\underline{\mathbf{a}} \cdot \underline{\mathbf{b}}}{|\underline{\mathbf{b}}|}$.
4. Find the projection of $\underline{\mathbf{b}}$ along $\underline{\mathbf{a}}$ using the formula $\frac{\underline{\mathbf{a}} \cdot \underline{\mathbf{b}}}{|\underline{\mathbf{a}}|}$.