# Thread: Findind a vector d

1. ## Findind a vector d

Let $\mathbf{\overrightarrow{a} = i + j -2k }$ , $\mathbf{\overrightarrow{b} = 2i + j + k}$ and $\mathbf{\overrightarrow{c} = 4i + j - 2k}$ . Find a vector $\mathbf{\overrightarrow{d}}$ such that $\mathbf{\overrightarrow{d} \times \overrightarrow{b} = \overrightarrow{c} \times \overrightarrow{b}}$ and $\mathbf{\overrightarrow{d} . \overrightarrow{a}}=0$.

2. ## Vector and scalar product

Hello varunnayudu
Originally Posted by varunnayudu
Let $\mathbf{\overrightarrow{a} = i + j -2k }$ , $\mathbf{\overrightarrow{b} = 2i + j + k}$ and $\mathbf{\overrightarrow{c} = 4i + j - 2k}$ . Find a vector $\mathbf{\overrightarrow{d}}$ such that $\mathbf{\overrightarrow{d} \times \overrightarrow{b} = \overrightarrow{c} \times \overrightarrow{b}}$ and $\mathbf{\overrightarrow{d} . \overrightarrow{a}}=0$.
Suppose that d = pi + qj + rk. Then equate the coefficients of i, j and k in vector products:

q - r = 3

2r - p = -8

p - 2q = 2

(Do you see where these equations come from?)

These equations are not all independent. You can't solve for p, q and r yet. But you also know that d.a = 0. So you also have:

p + q - 2r = 0

Now you can solve for p, q and r.