Question

Consider two vectors ${\overrightarrow{\mathbf{v}}}_{\mathbf{1}}$ and${\overrightarrow{\mathbf{v}}}_{\mathbf{2}}$in R3 that are not parallel.

Which vectors inlocalid="1668167992227" ${\mathbf{ℝ}}^{\mathbf{3}}$are linear combinations of${\overrightarrow{\mathbf{v}}}_{\mathbf{1}}$and${\overrightarrow{\mathbf{v}}}_{\mathbf{2}}$? Describe the set of these vectors geometrically. Include a sketch in your answer.

Short answer

The vectors of the form $c_1{\overrightarrow{v}}_1+c_2{\overrightarrow{v}}_2$a plane through the origin containing ${\overrightarrow{v}}_1$and ${\overrightarrow{v}}_2$.

Step-by-step solution

Step 1:

If vectors are not parallel then they are linearly independent. With the use parallelogram law of vector addition, we get that vector is linear combination of those 2 vectors if and only if it belongs to the same plane. Hence linear combination of the vectors ${\overrightarrow{v}}_1$ and ${\overrightarrow{v}}_2$ represents a plane i.e.

The vectors of the form $c_1{\overrightarrow{v}}_1+c_2{\overrightarrow{v}}_2$a plane through the origin containing ${\overrightarrow{v}}_1$ and ${\overrightarrow{v}}_2$.

Final answer:

The vectors of the form $c_1{\overrightarrow{v}}_1+c_2{\overrightarrow{v}}_2$ a plane through the origin containing ${\overrightarrow{v}}_1$and ${\overrightarrow{v}}_2$.