Subtracting Vectors with Three Components: Level 1 Example

Let’s subtract vector b = ⟨3, 5, 2⟩ from vector a = ⟨6, 4, 1⟩.

1. Write down vectors a and b: a = ⟨6, 4, 1⟩, b = ⟨3, 5, 2⟩.

2. Write down the negative of vector b: -b = ⟨-3, -5, -2⟩.

3. Add vector a and the negative of vector b: a – b = ⟨6, 4, 1⟩ + ⟨-3, -5, -2⟩.

4. Perform the addition component-wise: a – b = ⟨6 – 3, 4 – 5, 1 – 2⟩ = ⟨3, -1, -1⟩.

Conclusion: The result of subtracting vector b = ⟨3, 5, 2⟩ from vector a = ⟨6, 4, 1⟩ is ⟨3, -1, -1⟩. This process involves reversing the direction of the vector you want to subtract and then adding the vectors component-wise.

Example of subtracting vectors with three components: Given vectors c = ⟨4, 7, 9⟩ and d = ⟨2, 3, 1⟩, we perform the subtraction as follows: c – d = ⟨4, 7, 9⟩ – ⟨2, 3, 1⟩ = ⟨4, 7, 9⟩ + ⟨-2, -3, -1⟩ = ⟨4 – 2, 7 – 3, 9 – 1⟩ = ⟨2, 4, 8⟩. The result of subtracting vectors with three components c and d is ⟨2, 4, 8⟩.

When subtracting vectors with three components, such as vectors e = ⟨5, 3, 6⟩ and f = ⟨1, 2, 4⟩, the process is straightforward: e – f = ⟨5, 3, 6⟩ – ⟨1, 2, 4⟩ = ⟨5, 3, 6⟩ + ⟨-1, -2, -4⟩ = ⟨5 – 1, 3 – 2, 6 – 4⟩ = ⟨4, 1, 2⟩. The result of subtracting vectors with three components e and f is ⟨4, 1, 2⟩.

Example of subtracting vectors with three components: Given vectors \( g = \left\langle \frac{1}{2}, \frac{3}{4}, \frac{5}{6} \right\rangle \) and \( h = \left\langle \frac{1}{3}, \frac{2}{3}, \frac{1}{2} \right\rangle \), we perform the subtraction as follows:

\( g – h = \left\langle \frac{1}{2}, \frac{3}{4}, \frac{5}{6} \right\rangle – \left\langle \frac{1}{3}, \frac{2}{3}, \frac{1}{2} \right\rangle = \left\langle \frac{1}{2}, \frac{3}{4}, \frac{5}{6} \right\rangle + \left\langle -\frac{1}{3}, -\frac{2}{3}, -\frac{1}{2} \right\rangle \)

\( = \left\langle \frac{1}{2} – \frac{1}{3}, \frac{3}{4} – \frac{2}{3}, \frac{5}{6} – \frac{1}{2} \right\rangle = \left\langle \frac{3}{6} – \frac{2}{6}, \frac{6}{8} – \frac{4}{8}, \frac{5}{6} – \frac{3}{6} \right\rangle = \left\langle \frac{1}{6}, \frac{1}{4}, \frac{1}{3} \right\rangle \).

Conclusion: The result of subtracting vectors with three components \( g \) and \( h \) is \( \left\langle \frac{1}{6}, \frac{1}{4}, \frac{1}{3} \right\rangle \). This process involves reversing the direction of the vector you want to subtract and then adding the vectors component-wise, taking care to handle the fractions appropriately.

Example of subtracting vectors with three components: g – h = ⟨a, b, c⟩ – ⟨d, e, f⟩ = ⟨a – d, b – e, c – f⟩. This illustrates the process of subtracting vectors with three components.