Perform the indicated operations on the following vectors: ...

Perform the indicated operations on the following vectors: $$\begin{aligned} &\vec{a}=2 \vec{j}+\vec{k}, \quad \vec{b}=-3 \vec{i}+5 \vec{j}+4 \vec{k}, \quad \vec{c}=\vec{i}+6 \vec{j},\\ &\vec{x}=-2 \vec{i}+9 \vec{j}, \quad \vec{y}=4 \vec{i}-7 \vec{j}, \quad \vec{z}=\vec{i}-3 \vec{j}-\vec{k} \end{aligned}$$ $$4 \vec{z}$$

Perform the indicated operations on the following vectors:
$$\begin{aligned}
&\vec{a}=2 \vec{j}+\vec{k}, \quad \vec{b}=-3 \vec{i}+5 \vec{j}+4 \vec{k}, \quad \vec{c}=\vec{i}+6 \vec{j},\\
&\vec{x}=-2 \vec{i}+9 \vec{j}, \quad \vec{y}=4 \vec{i}-7 \vec{j}, \quad \vec{z}=\vec{i}-3 \vec{j}-\vec{k}
\end{aligned}$$
$$4 \vec{z}$$

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Key Concepts

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a constant (scalar), which scales the magnitude of the vector without changing its direction (unless the scalar is negative, which reverses the direction). This operation is performed by multiplying each component of the vector by the scalar.

Component Representation of Vectors

Vectors can be expressed in terms of their components along specified directions (typically along the i, j, and k unit vectors in three-dimensional space). This representation allows for straightforward calculation and manipulation of vectors since operations such as addition, subtraction, and scalar multiplication are performed component-wise.

Basis Vectors

Basis vectors, such as i, j, and k in three-dimensional space, are unit vectors that define the coordinate axes. They serve as the building blocks for constructing any vector in the space by scaling and combining these unit vectors. Understanding basis vectors is essential for interpreting vector component form and for performing vector operations.

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