If 𝐀=-12 𝐢̂+25 𝐣̂+13 𝐤̂ and 𝐁=-3 𝐣̂+7 𝐤̂, find the resultant...

If $\overrightarrow{\mathbf{A}}=-12 \hat{\mathbf{i}}+25 \hat{\mathbf{j}}+13 \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{B}}=-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}$, find the resultant when $\overrightarrow{\mathbf{A}}$ is subtracted from $\overrightarrow{\mathbf{B}}$. Erom a purely mathematical approach, $$ \begin{aligned} \overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}} &=(-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}})-(-12 \hat{\mathbf{i}}+25 \hat{\mathbf{j}}+13 \hat{\mathbf{k}}) \\ &=-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}+12 \hat{\mathbf{i}}-25 \hat{\mathbf{j}}-13 \hat{\mathbf{k}}=12 \hat{\mathbf{i}}-28 \hat{\mathbf{j}}-6 \hat{\mathbf{k}} \end{aligned} $$ Notice that $12 \hat{\mathbf{i}}-25 \hat{\mathbf{j}}-13 \hat{\mathbf{k}}$ is simply $\overrightarrow{\mathbf{A}}$ reversed in direction. Therefore, we have, in essence, reversed $\overrightarrow{\mathbf{A}}$ and added it to $\mathbf{B}$.

If $\overrightarrow{\mathbf{A}}=-12 \hat{\mathbf{i}}+25 \hat{\mathbf{j}}+13 \hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{B}}=-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}$, find the resultant when $\overrightarrow{\mathbf{A}}$ is subtracted from $\overrightarrow{\mathbf{B}}$.
Erom a purely mathematical approach,
$$
\begin{aligned}
\overrightarrow{\mathbf{B}}-\overrightarrow{\mathbf{A}} &=(-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}})-(-12 \hat{\mathbf{i}}+25 \hat{\mathbf{j}}+13 \hat{\mathbf{k}}) \\
&=-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}+12 \hat{\mathbf{i}}-25 \hat{\mathbf{j}}-13 \hat{\mathbf{k}}=12 \hat{\mathbf{i}}-28 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}
\end{aligned}
$$
Notice that $12 \hat{\mathbf{i}}-25 \hat{\mathbf{j}}-13 \hat{\mathbf{k}}$ is simply $\overrightarrow{\mathbf{A}}$ reversed in direction. Therefore, we have, in essence, reversed $\overrightarrow{\mathbf{A}}$ and added it to $\mathbf{B}$.

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Key Concepts

Vector Subtraction

Vector subtraction is an operation that finds the difference between two vectors by subtracting their corresponding components. This process produces a new vector that represents the displacement from one vector to the other and is computed by performing component-wise subtraction, i.e., subtracting the coefficients of each unit vector (i, j, k).

Component Form Representation

Expressing vectors in component form using unit vectors such as i, j, and k allows for clear and systematic handling of vector operations. This representation breaks the vector into its individual directional components, simplifying addition, subtraction, and other vector operations by treating each dimension separately.

Vector Negation

Vector negation involves reversing the direction of a vector by multiplying it by -1, which changes the sign of each component. This concept is fundamental in vector operations because subtracting a vector is equivalent to adding its negative, thus simplifying the operation into an addition of the reversed vector.

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