Science:Math Exam Resources/Courses/MATH152/April 2022/Question A11
• QA01 • QA02 • QA03 • QA04 • QA05 • QA06 • QA07 • QA08 • QA09 • QA10 • QA11 • QA12 • QA13 • QA14 • QA15 • QA16 • QA17 • QA18 • QA19 • QA20 • QB1 (a) • QB1 (b) • QB2 (a) • QB2 (b) • QB2 (c) • QB3 (a) • QB3 (b) • QB4 (a) • QB4 (b) • QB4 (c) • QB5 (a) • QB5 (b) • QB5 (c) •
Question A11 

Let be planes in 3space defined by equations Find the parametric form of the line of intersection of these planes. 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Recall that the parametric form of the line is . To solve this problem you need to find the point and the vector . Can you use what you know about the normal vectors of planes to determine the direction vector of the line of intersection? 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 1 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. The line of intersection of planes and will be perpendicular to both (the normal vector of ) and (the normal vector of ). Since the cross product of two linearly independent vectors in results in a vector that is perpendicular to both of them, we will take the cross product of and to get the direction vector for the line of intersection. To find the point we solve the system of equations using row operations. We now have the system of equations Rearranging and solving them for and gives: and . Since is a free variable now, we can choose and substitute this into the expressions for and to determine the point . We have . Concluding that the line is 
Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Alternatively, this problem can be solved simply by solving the system equations we solved to find the point in the previous solution. To recap, that system is Because this is a system with more variables than equations, it will either have no solution, or infinitely many solutions. Looking at the normal vectors for the planes, we can see that they are not parallel, meaning that the planes will intersect along a line! Recall that the system simplified to Now, since is a free variable, let , giving the system 