Suppose a = ⟨3, 1, 1⟩, b = ⟨2, −1, 1⟩, and c = ⟨0, 1, 2⟩. Obtain a vector which is
coplanar with a and b

To find a vector that is coplanar with vectors \(\mathbf{a} = \langle 3, 1, 1 \rangle\) and \(\mathbf{b} = \langle 2, -1, 1 \rangle\), we can take a linear combination of \(\mathbf{a}\) and \(\mathbf{b}\). This means any vector \(\mathbf{v}\) that is coplanar with \(\mathbf{a}\) and \(\mathbf{b}\) can be expressed as:

\[
\mathbf{v} = \alpha \mathbf{a} + \beta \mathbf{b}
\]

where \(\alpha\) and \(\beta\) are scalars.

Let's choose arbitrary values for \(\alpha\) and \(\beta\) to obtain a specific coplanar vector.

For simplicity, let's choose \(\alpha = 1\) and \(\beta = 1\):

\[
\mathbf{v} = 1 \cdot \mathbf{a} + 1 \cdot \mathbf{b}
\]

Substituting \(\mathbf{a}\) and \(\mathbf{b}\):

\[
\mathbf{v} = \langle 3, 1, 1 \rangle + \langle 2, -1, 1 \rangle
\]

Adding the vectors component-wise:

\[
\mathbf{v} = \langle 3 + 2, 1 + (-1), 1 + 1 \rangle = \langle 5, 0, 2 \rangle
\]

Therefore, one vector that is coplanar with \(\mathbf{a}\) and \(\mathbf{b}\) is:

\[
\mathbf{v} = \langle 5, 0, 2 \rangle
\]

Would you like more details or have any questions? Here are some additional questions you might be interested in:

1. How to verify if three vectors are coplanar?
2. What is the cross product of two vectors?
3. How to find the area of the parallelogram formed by two vectors?
4. How to find the volume of the parallelepiped formed by three vectors?
5. How to determine if a vector is perpendicular to a plane?

**Tip:** When dealing with vector operations, always keep track of the components and the rules for vector addition, scalar multiplication, and other vector operations.

Learn how to find a vector that is coplanar with vectors a = ⟨3, 1, 1⟩ and b = ⟨2, −1, 1⟩. Follow a step-by-step solution using vector addition and scalar multiplication methods. Suitable for high school students and anyone interested in vector mathematics.

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