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If a = 2i + 3j + 4k and b = 3i − 2j + 1k, determine (a) a + b, (b) a - b, (c) a + 2b.

Mathematics

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Answers

3 parts
a

5i⃗+1j⃗+5k⃗5\vec{i} + 1\vec{j} + 5\vec{k}

b

−1i⃗+5j⃗+3k⃗-1\vec{i} + 5\vec{j} + 3\vec{k}

c

8i⃗−1j⃗+6k⃗8\vec{i} - 1\vec{j} + 6\vec{k}

Step-by-step solution

10 steps

Let's break down these vector operations.

Given vectors:
a⃗=2i⃗+3j⃗+4k⃗\vec{a} = 2\vec{i} + 3\vec{j} + 4\vec{k}
b⃗=3i⃗−2j⃗+1k⃗\vec{b} = 3\vec{i} - 2\vec{j} + 1\vec{k}

Part a· 3 steps

Determine a⃗+b⃗\vec{a} + \vec{b}

  1. Write out the vectors for addition.

    a⃗+b⃗=(2i⃗+3j⃗+4k⃗)+(3i⃗−2j⃗+1k⃗)\vec{a} + \vec{b} = (2\vec{i} + 3\vec{j} + 4\vec{k}) + (3\vec{i} - 2\vec{j} + 1\vec{k})

  2. Group the corresponding components (i⃗\vec{i} with i⃗\vec{i}, j⃗\vec{j} with j⃗\vec{j}, k⃗\vec{k} with k⃗\vec{k}).

    a⃗+b⃗=(2+3)i⃗+(3−2)j⃗+(4+1)k⃗\vec{a} + \vec{b} = (2+3)\vec{i} + (3-2)\vec{j} + (4+1)\vec{k}

  3. Perform the addition for each component.

    a⃗+b⃗=5i⃗+1j⃗+5k⃗\vec{a} + \vec{b} = \boxed{5\vec{i} + 1\vec{j} + 5\vec{k}}

Part b

−1i⃗+5j⃗+3k⃗-1\vec{i} + 5\vec{j} + 3\vec{k}

Determine a⃗−b⃗\vec{a} - \vec{b}

  1. Write out the vectors for subtraction.

    a⃗−b⃗=(2i⃗+3j⃗+4k⃗)−(3i⃗−2j⃗+1k⃗)\vec{a} - \vec{b} = (2\vec{i} + 3\vec{j} + 4\vec{k}) - (3\vec{i} - 2\vec{j} + 1\vec{k})

  2. Group the corresponding components, remembering to distribute the negative sign.

    a⃗−b⃗=(2−3)i⃗+(3−(−2))j⃗+(4−1)k⃗\vec{a} - \vec{b} = (2-3)\vec{i} + (3-(-2))\vec{j} + (4-1)\vec{k}

  3. Perform the subtraction for each component.

    a⃗−b⃗=−1i⃗+5j⃗+3k⃗\vec{a} - \vec{b} = \boxed{-1\vec{i} + 5\vec{j} + 3\vec{k}}

Part c

8i⃗−1j⃗+6k⃗8\vec{i} - 1\vec{j} + 6\vec{k}

Determine a⃗+2b⃗\vec{a} + 2\vec{b}

  1. First, calculate 2b⃗2\vec{b} by multiplying each component of b⃗\vec{b} by 2.

    2b⃗=2(3i⃗−2j⃗+1k⃗)=6i⃗−4j⃗+2k⃗2\vec{b} = 2(3\vec{i} - 2\vec{j} + 1\vec{k}) = 6\vec{i} - 4\vec{j} + 2\vec{k}

  2. Now, add a⃗\vec{a} and 2b⃗2\vec{b}.

    a⃗+2b⃗=(2i⃗+3j⃗+4k⃗)+(6i⃗−4j⃗+2k⃗)\vec{a} + 2\vec{b} = (2\vec{i} + 3\vec{j} + 4\vec{k}) + (6\vec{i} - 4\vec{j} + 2\vec{k})

  3. Group the corresponding components.

    a⃗+2b⃗=(2+6)i⃗+(3−4)j⃗+(4+2)k⃗\vec{a} + 2\vec{b} = (2+6)\vec{i} + (3-4)\vec{j} + (4+2)\vec{k}

  4. Perform the addition for each component.

    a⃗+2b⃗=8i⃗−1j⃗+6k⃗\vec{a} + 2\vec{b} = \boxed{8\vec{i} - 1\vec{j} + 6\vec{k}}
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