### Video Transcript

Find, to the nearest second, the angle between the straight line ๐ฅ equals three ๐ก minus one, ๐ฆ equals negative two ๐ก plus four, ๐ง equals five and the plane three ๐ฅ minus four ๐ฆ plus ๐ง equals two.

Okay, so here weโre given the equation of a straight line. And itโs given to us in whatโs called parametric form, written as three separate equations: one for ๐ฅ, one for ๐ฆ, and one for ๐ง. Weโre also given the equation of a plane. And this is nearly in the form thatโs called general form.

If we were to subtract two from both sides, we would get this result, which is indeed known as the general form of the equation of this plane. So we have our plane in general form and our line written in parametric form. Having all this, what we want to do first is solve for a vector that is parallel to our line and second for a vector that is normal or perpendicular to our plane.

If we call a vector in general that is parallel to a line ๐ฉ and a vector that is normal to a plane ๐ง, then the sine of the angle between that line and that plane is given by this expression. This is why we want to solve for a vector thatโs parallel to our given line and a vector thatโs normal to our given plane.

Coming back to the equation of our line in parametric form, thereโs a way that we can combine all three of these equations into one. We could say that the ๐ฅ-, ๐ฆ-, and ๐ง-values of this line are represented by a vector ๐ซ. Furthermore, the point negative one, four, five lies on the line, and the vector three, negative two, zero is parallel to it. This means that weโve solved for a vector ๐ฉ thatโs parallel to our line. It has components three, negative two, zero.

Next, to find the vector ๐ง that is normal to our plane, we can recall that since our plane is in general form, the values by which we multiply ๐ฅ, ๐ฆ, and ๐ง are the components of a vector normal to this plane. In other words, one vector normal to the plane has components three, negative four, one.

Now that we have our parallel and normal vectors, we can substitute them into this expression to ultimately solve for ๐. That gives us this expression, where in our numerator we have a dot product to calculate and in our denominator two vector magnitudes to compute. As we compute and simplify this fraction, we get to a point of having 17 divided by the square root of 13 times 26. This, we recall, is the sin of ๐. So to solve for ๐ itself, we take the inverse sine of both sides.

Entering this expression into our calculator and rounding to the nearest second, our result is 67 degrees, 37 minutes, and 12 seconds. This is the angle between our line and plane to the nearest second.