Vector problems!

Could somebody please help me with this question!? I have tried and tribe but am getting nowhere

I think I got part i, but have no clue about ii, or iii!

Any help is greatly appreciated!


Posted from TSR Mobile

Could somebody please help me with this question!? I have tried and tribe but am getting nowhere

I think I got part i, but have no clue about ii, or iii!

Any help is greatly appreciated!


Posted from TSR Mobile

Part (i) is correct.

For part (ii) you need to form a vector equation of the form $\textbf{r}=\textbf{c}+\lambda \textbf{d}$ where $\textbf{c}$ is any point on the line, and $\textbf{d}$ is a direction on the line. You are told that the line passes through A and B, so you can use either of those as your points. But now you need to find a direction on the line... given that you know the line goes through A and B, can you think of what direction vector you would want to use and how you would go about calculating it?

For (iii) you'll want to equate your finished vector equation to the point (5,-7-4). By reading off the rows, you can form three equations. You will then show that the value of lambda is the same in each of the three equations, this proves that the entire vector equation itself is consistent and means that the point lies on the line.

Let us know how you get on.

Ok well I don't know how to find the direction vector or how to calculate it? Also, for part i) should i use SQRT((3-1)^2 + (-2-3)^2 + (0-4)^2) or what I used? I know they give the same answer, but which is the correct way?

Ok this is what I got for ii)
Also I'm very confused by iii

Posted from TSR Mobile

Yep, that looks correct to me. It's always useful for these types of questions to remember that the direction vector between two points can always be found by subtracting the end point from the start point, like you have done here

For the final question, I think it might be a little bit easier to visualise if we rewrite this in column vector form.

So, taking your answer and rewriting it, we have: $\textbf{r}=\begin{pmatrix}1 \\ 3 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix}2 \\ -5 \\ -4 \end{pmatrix}$

Recall that $\textbf{r}$ simply represents the column vector of the variables, or in other words, we can write $\begin{pmatrix}x \\ y \\ z \end{pmatrix}=\begin{pmatrix}1 \\ 3 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix}2 \\ -5 \\ -4 \end{pmatrix}$

The task now is to simply substitute $\textbf{r}$ with the point that we are trying to prove lies on the line, $\begin{pmatrix}2 \\ -5 \\ -4 \end{pmatrix}$.

So we write $\begin{pmatrix}2 \\ -5 \\ -4 \end{pmatrix} = \begin{pmatrix}1 \\ 3 \\ 4 \end{pmatrix} + \lambda \begin{pmatrix}2 \\ -5 \\ -4 \end{pmatrix}$

We want to show that this equation is consistent, that everything holds and that it all works out without any inconsistencies or contradictions arising. In other words, we want to show that there exists a value of $\lambda$ such that the above equation works. Can you think of a way of doing this?

That point does not lie on the line and it is not the point we are trying to show does lie on the line.