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Q54E

Expert-verifiedFound in: Page 37

Book edition
5th

Author(s)
David C. Lay, Steven R. Lay and Judi J. McDonald

Pages
483 pages

ISBN
978-03219822384

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

**Which vectors in localid="1668167992227" ${{\mathbf{\mathbb{R}}}}^{{\mathbf{3}}}$ 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.**

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 ${\overrightarrow{v}}_{2}$.

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 ${\overrightarrow{v}}_{2}$.*

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 ${\overrightarrow{v}}_{2}$ .

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