To determine which vector is orthogonal to $\mathbf{u} = (1,7)$, we need to find a vector $\mathbf{v} = (a, b)$ such that their dot product is zero.

The dot product of two vectors $\mathbf{u} = (u_1, u_2)$ and $\mathbf{v} = (v_1, v_2)$ is calculated as: $\mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2$ For orthogonality, this product should equal zero.

Given $\mathbf{u} = (1,7)$, we calculate the dot product with each choice:

1. $(1, -7)$: $(1)(1) + (7)(-7) = 1 - 49 = -48$ Not zero, so this vector is not orthogonal to $\mathbf{u}$.

2. $(-1, 7)$: $(1)(-1) + (7)(7) = -1 + 49 = 48$ Not zero, so this vector is not orthogonal to $\mathbf{u}$.

3. $(7, -1)$: $(1)(7) + (7)(-1) = 7 - 7 = 0$ This equals zero, so $(7, -1)$ is orthogonal to $\mathbf{u}$.

4. $(7, 1)$: $(1)(7) + (7)(1) = 7 + 7 = 14$ Not zero, so this vector is not orthogonal to $\mathbf{u}$.

Answer

The vector $(7, -1)$ is orthogonal to $\mathbf{u} = (1,7)$.

Would you like a deeper explanation or further examples?

Here are five related questions to explore:

1. How can we determine if two vectors in three-dimensional space are orthogonal?
2. What properties do orthogonal vectors have in the context of vector spaces?
3. How would you find a vector orthogonal to a given vector in three-dimensional space?
4. If two vectors are orthogonal, what can be said about the angle between them?
5. How does the dot product relate to the cosine of the angle between two vectors?

Tip: If two vectors are orthogonal, their dot product will always be zero. This is a quick way to check orthogonality.