Short Answer

Consider two vectors ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ in R3 that are not parallel.
Which vectors in localid="1668167992227" $ℝ^{3}$ are linear combinations of ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ ? Describe the set of these vectors geometrically. Include a sketch in your answer.

The vectors of the form $c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2}$ a plane through the origin containing ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ represents a plane i.e.

The vectors of the form $c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2}$ a plane through the origin containing ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ .

Step 2: Final answer:

The vectors of the form $c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2}$ a plane through the origin containing ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ .

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Linear Algebra and its Applications

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Short Answer

Consider two vectors ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ in R3 that are not parallel.
Which vectors in localid="1668167992227" $ℝ^{3}$ are linear combinations of ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ ? Describe the set of these vectors geometrically. Include a sketch in your answer.

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Linear Algebra and its Applications

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Consider two vectors v→1 and v→2 in R3 that are not parallel.Which vectors in localid="1668167992227" ℝ3 are linear combinations of v→1 and v→2? Describe the set of these vectors geometrically. Include a sketch in your answer.

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Consider two vectors ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ in R3 that are not parallel.
Which vectors in localid="1668167992227" $ℝ^{3}$ are linear combinations of ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ ? Describe the set of these vectors geometrically. Include a sketch in your answer.