Vector 𝐀 has x and y components of -8.70 cm and 15.0 ...

Vector $\overrightarrow{\mathbf{A}}$ has $x$ and $y$ components of $-8.70 \mathrm{~cm}$ and $\begin{array}{llllll}15.0 & \mathrm{~cm}, & \text { respectively; } & \text { vector } & \overrightarrow{\mathbf{B}} & \text { has } & x & \text { and } & y & \operatorname{com}-\end{array}$ $\begin{array}{llllll}\text { ponents of } 13.2 & \mathrm{~cm} & \text { and } & -6.60 & \mathrm{~cm}, & \text { respectively. }\end{array}$ If $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}+3 \overrightarrow{\mathbf{C}}=0,$ what are the components of $\overrightarrow{\mathbf{C}} ?$

Vector $\overrightarrow{\mathbf{A}}$ has $x$ and $y$ components of $-8.70 \mathrm{~cm}$ and $\begin{array}{llllll}15.0 & \mathrm{~cm}, & \text { respectively; } & \text { vector } & \overrightarrow{\mathbf{B}} & \text { has } & x & \text { and } & y & \operatorname{com}-\end{array}$
$\begin{array}{llllll}\text { ponents of } 13.2 & \mathrm{~cm} & \text { and } & -6.60 & \mathrm{~cm}, & \text { respectively. }\end{array}$
If $\overrightarrow{\mathbf{A}}-\overrightarrow{\mathbf{B}}+3 \overrightarrow{\mathbf{C}}=0,$ what are the components of $\overrightarrow{\mathbf{C}} ?$

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Key Concepts

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a constant, which scales the vector’s magnitude without altering its direction. Each component of the vector is multiplied by the scalar, uniformly stretching or shrinking the vector.

Solving Vector Equations

When solving vector equations, the equality of vectors is applied component by component. This results in a system of equations for each dimension, which can then be solved using standard algebraic techniques to find unknown vector components.

Vector Representation in Cartesian Coordinates

This concept involves expressing a vector in terms of its horizontal (x) and vertical (y) components. It allows complex vector quantities to be broken down into simpler, more manageable parts, making subsequent operations like addition or subtraction straightforward.

Component-wise Vector Operations

Vector operations such as addition and subtraction are performed by operating on each corresponding component individually. This method of handling vectors simplifies problem-solving by reducing multi-dimensional vector operations to basic arithmetic with scalars.

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