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To find vectors perpendicular to the given vectors, we need to use the concept of the dot product. For two vectors to be perpendicular, their dot product must equal zero.

### Part (a) - Vector \(\langle -1, 3 \rangle\)
Let the perpendicular vector be \(\langle x, y \rangle\).

We need to solve the following equation:

\[
(-1) \cdot x + 3 \cdot y = 0
\]

This simplifies to:

\[
-x + 3y = 0 \quad \Rightarrow \quad x = 3y
\]

Thus, any vector of the form \(\langle 3y, y \rangle\) is perpendicular to \(\langle -1, 3 \rangle\). Since \(y\) can take any real value, there are **infinitely many vectors** that are perpendicular.

### Part (b) - Vector \(\langle 2, -1, 4 \rangle\)
Let the perpendicular vector be \(\langle x, y, z \rangle\).

We need to solve the following equation:

\[
2 \cdot x + (-1) \cdot y + 4 \cdot z = 0
\]

This simplifies to:

\[
2x - y + 4z = 0
\]

This equation has two degrees of freedom. Thus, there are **infinitely many solutions** since \(x\), \(y\), and \(z\) can vary while satisfying the equation.

### Conclusion:
For both vectors (a and b), there are infinitely many answers.

Would you like a more detailed explanation or any clarifications?

Here are five questions that expand on this information:
1. How can you derive a general form for vectors perpendicular to a given vector in 2D?
2. What is the geometric interpretation of vectors in 3D being perpendicular?
3. How can you check if two vectors are orthogonal in higher dimensions?
4. How would you solve for a perpendicular vector if more than two vectors are involved?
5. Can the cross product be used to find perpendicular vectors in 3D?

**Tip**: Always check the dot product when determining if two vectors are perpendicular; it should equal zero for orthogonality.

Find a vector perpendicular to each of the vectors. How many answers are there? a. <-1, 3> b. <2, -1, 4>

Learn how to find vectors perpendicular to the vectors <-1, 3> and <2, -1, 4> using the dot product. This problem explores key concepts of linear algebra, including the orthogonality theorem and dot product calculations. Suitable for college-level students studying linear algebra.

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