Science:Math Exam Resources/Courses/MATH105/April 2016/Question 01 (a)

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MATH105 April 2016

  •  Q1 (a)  •  Q1 (b)  •  Q1 (c)  •  Q1 (d)  •  Q1 (e)  •  Q1 (f)  •  Q1 (g)  •  Q1 (h)  •  Q1 (i)  •  Q1 (j)  •  Q1 (k)  •  Q1 (l)  •  Q1 (m)  •  Q1 (n)  •  Q2 (a)  •  Q2 (b)  •  Q2 (c)  •  Q2 (d)  •  Q3  •  Q4 (a)  •  Q4 (b)  •  Q5 (a)  •  Q5 (b)  •  Q5 (c)  •  Q5 (d)  •  Q6  •  

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Question 01 (a)
Find an equation of the plane that passes through the point $(-1,0,3)$ with a normal vector $\langle 3,5,-2\rangle$.

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
What is the equation of plane when the normal vector and a point on the plane are given?

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution 1

An equation of a plane that passes through the point $P=(x_{0},y_{0},z_{0})$ and is orthogonal to a vector $n=\langle a,b,c\rangle$ (a normal vector) is given by $a(x-x_{0})+b(y-y_{0})+c(z-z_{0})=0$ or by the simplified form $ax+by+cz=d,$ where $d=ax_{0}+by_{0}+cz_{0}$ . Here, we have the point $P=(-1,0,3)$ and the normal vector $n=\langle 3,5,-2\rangle$. So, the equation of the plane is $3(x-(-1))+5(y-0)+(-2)(z-3)=0$, which implies $3(x+1)+5y-2z+6=0$ . We then get ${\begin{aligned}3x+5y-2z=-9.\end{aligned}}$

Solution 2
Let $(x,y,z)$ be a point on the plane. Then the vector $(x,y,z)-(-1,0,3)=(x+1,y,z-3)$ lies on the plane. Since a normal vector on a plane is orthogonal to any vector lying on the plane, we have ${\begin{aligned}0=(x+1,y,z-3)\cdot (3,5,-2)=3(x+1)+5y-2(z-3)=3x+5y-2z+9\end{aligned}}$. Therefore, the equation for the plane is $3x+5y-2z=-9$.

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