I'm busy studying for my Physics exam this evening and I've come across a vector problem that I cant quite solve. Any help would be appreciated!

We are given the following vectors:

A = 5i - 6.5j
B = -3.5i + 7j

We are also told that a vector C lies in the xy-plane and is perpendicular to A. The scalar product of B and C is equal to 15. With this information we have to calculate the components of the vector C.

I have tried calculating the magnitude of B and then trying to solve the equation:

|B||C|cos{theta} = 15

But I didn't get very far. I have a feeling I'm just not seeing something and making a careless error.

Thanks in advance!

  • 1
    $\begingroup$ Let C = $x$i + $y$j. If A and C are perpendicular you know A.C = 0, and you're told B.C = 15. If you multiply out the dot products you get two simultaneous equations for $x$ and $y$. $\endgroup$ Commented May 21, 2013 at 10:15
  • $\begingroup$ That makes a lot more sense! Thank you! I tried this on paper now and I got C = -1.69i + 1.29j. Is this right? $\endgroup$
    – Nick Corin
    Commented May 21, 2013 at 10:35
  • $\begingroup$ @nickcorin: I get another result: $C=\frac{390}{49}i+\frac{300}{49}j$. Which equations do you get when working out A.C=0 and B.C=15? $\endgroup$
    – fibonatic
    Commented May 21, 2013 at 10:58
  • $\begingroup$ @fibonatic: we're not supposed to give the answers to homework questions, which is why I specifically avoided doing so. $\endgroup$ Commented May 21, 2013 at 11:00
  • $\begingroup$ @nickcorin: have you tried substituting the answers you got into the original equations to check they are correct? $\endgroup$ Commented May 21, 2013 at 11:01

2 Answers 2


Here is another approach that is essentially the same as the one suggested by John, but I find it to be less error-prone when doing calculations by hand.

You are given

$$ \mathbf{A} = 5\mathbf{i} - 6.5\mathbf{j}\\ \mathbf{B} = -3.5\mathbf{i} + 7\mathbf{j} $$

You know that $ \mathbf{C} \perp \mathbf{A}$. Therefore,

$$ \mathbf{C} = s \cdot (6.5 \mathbf{i} + 5\mathbf{j}) $$

with $s$ a scale factor. This is because in 2D, for any vector $\alpha = a\mathbf{i} + b\mathbf{j}$, the vector $\beta = -b\mathbf{i} + a\mathbf{j}$ satisfies $\alpha \cdot \beta = 0$ and thus $\alpha \perp \beta$.

Given that $\mathbf{B} \cdot \mathbf{C} = 15$, it should be fairly easy to solve for the scale factor $s$, thus solving for $\mathbf{C}$.

  • $\begingroup$ This makes a lot more sense to me, although I can't see where I should go from here. $\endgroup$
    – Nick Corin
    Commented May 21, 2013 at 12:31
  • $\begingroup$ @nickcorin: You know that $\mathbf{C}\cdot\mathbf{B}=15$. Just substitute the expression for $\mathbf{C}$ above and solve for $s$. $\endgroup$ Commented May 21, 2013 at 14:14

Convert it into a simultaneous equation

Let C = xi + yj

we know that A = 5i - 6.5j B = -3.5i + 7j C.B = 15 C.A = 0 (because the magnitude of perpendicular vectors is 0)

C.A --> 5x - 6.5y = 0

C.B --> -3.5x + 7y = 15


  • 17.5x + 22.75y = 0
  • 17.5x + 35y = 15 y = 60/49 x = 78/49

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.