Let L1 be the line with the vector equation r= (-2,3) +s(-1,3), seR. Let L2 be with the line with parametric equation x=3t+2, y=t-7, teR. Are L1 and L2 perpendicular? How do you know?

You have to look at their direction numbers

direction of L1 is (-1,3)

for L2, the direction is (3,1)

since (3,1)(-1,3) = 0 , they are perpendicular.

To determine if two lines are perpendicular, we can check if the direction vectors of the lines are orthogonal, meaning their dot product is zero.

First, we need to find the direction vector for each line.

For L1, the vector equation is given as r = (-2,3) + s(-1,3).

- The direction vector for L1 is the coefficient of s, which is (-1,3).

For L2, the parametric equation is given as x = 3t + 2 and y = t - 7.

- To find the direction vector for L2, we take the coefficients of t, which are (3,1).

Now, we can calculate the dot product of these two direction vectors. The dot product of two vectors A = (a1, a2) and B = (b1, b2) is given by:

A · B = (a1 * b1) + (a2 * b2).

Using this formula, we calculate the dot product of the direction vectors:

(-1,3) · (3,1) = (-1 * 3) + (3 * 1) = -3 + 3 = 0.

Since the dot product of the direction vectors is zero, we conclude that L1 and L2 are perpendicular.