# Thread: Solve my vector Problems

1. ## Solve my vector Problems

a) Show that the points (1,3,1) ,(1,1,-1),(-1,1,1) (2,2,-1) are lying on the same plane

b) For any vector r prove that r = (r.i)i+(r.j)j+(r.k)k

c) If a*(b*c)=(a*b)*c then prove that (c*a)*b=0

2. Originally Posted by Salman91
a) Show that the points (1,3,1) ,(1,1,-1),(-1,1,1) (2,2,-1) are lying on the same plane
Let P = (1,3,1), Q = (1,1,-1), R = (-1,1,1) and S = (2,2,-1)

Show that
$\displaystyle \|\vec {PQ} \cdot (\vec{PR} \times \vec{PS}) \| = 0$

b) For any vector r prove that r = (r.i)i+(r.j)j+(r.k)k
$\displaystyle \vec r = \left< r_i, r_j, r_k \right> = \left< r_i, 0, 0 \right> + \left< 0, r_j, 0 \right> + \left< 0, 0, r_k \right> = \cdots$

think you can finish up?

c) If a*(b*c)=(a*b)*c then prove that (c*a)*b=0
what are you using * to mean?

3. * = multiply ( x )

4. Originally Posted by Salman91
* = multiply ( x )
i assume that a, b, c are vectors here. "multiply" makes no sense. do you mean "cross product"? and they are all cross products? you have to distinguish between the dot and cross product!

5. yes sir it is a cross-product

and can you please show me the answer of part b (step by step)

6. Originally Posted by Salman91
yes sir it is a cross-product
i will just solve this one...

recall these properties from your text: $\displaystyle a \times ( b \times c) = (a \cdot c)b - (a \cdot b)c$, $\displaystyle a \times b = -b \times a$ and $\displaystyle a \cdot b = b \cdot a$ for any vectors $\displaystyle a,b,c$ (of course, for cross-products we need 3-d vectors)

Note that $\displaystyle a \times (b \times c) = (a \cdot c)b - (a \cdot b)c$

$\displaystyle (a \times b) \times c = -c \times (a \times b) = (-c \cdot b)a - (-c \cdot a)b$ and

$\displaystyle (c \times a) \times b = -b \times (c \times a) = (-b \cdot a)c - (-b \cdot c)a$

So, assume $\displaystyle a \times (b \times c) = (a \times b) \times c$, then

$\displaystyle (a \cdot c)b - (a \cdot b)c = (-c \cdot b)a - (-c \cdot a)b$

$\displaystyle \Rightarrow (a \cdot c)b - (a \cdot b)c - (-c \cdot b)a + (-c \cdot a)b = 0$

$\displaystyle \Rightarrow (a \cdot c)b + (-b \cdot a)c - (-b \cdot c)a - (a \cdot c)b = 0$

$\displaystyle \Rightarrow (-b \cdot a)c - (-b \cdot c)a = 0$

$\displaystyle \Rightarrow (c \times a) \times b = 0$
and can you please show me the answer of part b (step by step)
there is pretty much one step left, you can't finish it? try.

### for any vector r prove tha r.i

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