Question 9 of 68beginner🔧 ApplyNumerical1 mark

Find the sum of the vectors a⃗ = î - 2ĵ + k̂, b⃗ = -2î + 4ĵ + 5k̂ and c⃗ = î - 6ĵ - 7k̂.

Correct Answer

-4j^−k^\hat{j} - \hat{k}

Exercise: EXERCISE 10.2 | Q: 6 | (Page 17)
For More Understanding

Explanation

To find the sum of the given vectors, we use the property that the sum of vectors is obtained by adding their corresponding components. Specifically, the coefficient of i^\hat{i} in the resultant vector is the sum of the coefficients of i^\hat{i} from all vectors, and similarly for j^\hat{j} and k^\hat{k}.

Solution Steps

  1. Step 1: Identify the components of each vector: a⃗=i^−2j^+k^\vec{a} = \hat{i} - 2\hat{j} + \hat{k}, b⃗=−2i^+4j^+5k^\vec{b} = -2\hat{i} + 4\hat{j} + 5\hat{k}, and c⃗=i^−6j^−7k^\vec{c} = \hat{i} - 6\hat{j} - 7\hat{k}.

  2. Step 2: Calculate the sum of the coefficients of i^\hat{i}: 1+(−2)+1=01 + (-2) + 1 = 0.

  3. Step 3: Calculate the sum of the coefficients of j^\hat{j}: (−2)+4+(−6)=−4(-2) + 4 + (-6) = -4.

  4. Step 4: Calculate the sum of the coefficients of k^\hat{k}: 1+5+(−7)=−11 + 5 + (-7) = -1.

  5. Step 5: Combine the results to form the resultant vector: 0i^−4j^−1k^=−4j^−k^0\hat{i} - 4\hat{j} - 1\hat{k} = -4\hat{j} - \hat{k}.

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