### Calculus in Three Dimensions (21-259) — Feedback on Homework 2

Homework 2 was due on Tuesday 9th September 2014. Questions 1, 2, 3, 4 and 5 were graded.

**Question 1.** The most common error in part (a) was people simply writing the vector $\overrightarrow{AB}$ for the equation of the line. This is the *direction* of the line; the vector equation of the line is $\mathbf{r}(t) = \overrightarrow{OA} + t\overrightarrow{AB}$.

For part (b), the vector equation of a line segment is the same as the vector equation of a line, except the values of $t$ are restricted. The line segment from $A$ to $B$ is thus the same as in part (a), but with a restriction on $t$.

A common problem with part (c) was people taking the direction vector of the line as being the normal vector to the plane. This isn't true: the line lies in the plane, so in fact the plane is parallel to the direction vector of the line! You needed to use the fact that, say, both $\overrightarrow{AB}$ and $\overrightarrow{AC}$ lie in the plane, and take their cross product to find the normal vector.

**Question 2.** Most people who got this question wrong did so because they misunderstood what 'coordinate plane' meant. You had to find the values of $t$ for which $x=0$, $y=0$ or $z=0$, and then find the corresponding points on the line.

**Question 3.** As with 1(c), the main error here was taking the direction vector of the line as being the normal vector of the plane.

**Question 4.** This question was mostly done very well. Those who messed it up usually did so because they simply set the $x$, $y$ and $z$ coordinates equal to each other without renaming one of the parameters. The idea is: as the parameters vary, the vector traces out a line; the lines intersect, but the value of the parameters on the two lines at the point of intersection aren't the same.

**Question 5.** If you made it all the way to the end of this fiddly question then I commend you. It was done surprisingly well, with the most common error being people mixing up their $-$ signs when taking a cross product.

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