Find the coordinates of the following vectors with respect to the basis

Question

The vectors v1 = ($\frac{3}{7}$, $\frac{6}{7}$,$\frac{−2}{7}$), v2 = ($\frac{−2}{7}$, $\frac{3}{7}$, $\frac{6}{7}$) and v3 = ($\frac{6}{7}$,$\frac{−2}{7}$, $\frac{3}{7}$) form an orthonormal basis of ℝ$^3$. Find the coordinates of the following vectors with respect to this basis.

(i) (1,0,0)

(ii) (2, N+10, 1)

I am wondering is there anyway I can find the coordinates without using reduced row form? For vectors in ℝ$^2$ you can use the dot product and a formula but I am not sure if that works for vectors in ℝ$^3$?

Any advice would be greatly appreciated.

Answer 1

If $v\in\Bbb R^3$, then$$v=(v.v_1)v_1+(v.v_2)v_2+(v.v_3)v_3.$$So, since, for instance,$$(1,0,0).v_1=\frac37,\ (1,0,0).v_2=-\frac27\text{, and }(1,0,0).v_3=\frac67,$$you have$$(1,0,0)=\frac37v_1-\frac27v_2+\frac67v_3.$$

Answer 2

Letting $$ v=a\,v_1 + b\,v_2 +c\, v_3 $$ the co-ordinates of $v$ in the given basis are $(a,b,c)$.

Use the orthogonality of the basis: $$ v_1 \cdot v =a\,v_1 \cdot v_1 + b 0+c 0=a\,v_1 \cdot v_1 $$ and use the normalised property of the basis $v_1 \cdot v_1=1$ $$ v_1 \cdot v =a $$ Find the other co-ordinates similarly.

If the basis is not orthogonal you can still generate a vector normal to $v_2$ and $v_3$ using the vector product $v_2 \wedge v_3$ and then use that to take scalar products to find the co-ordinates e.g. $$ v \cdot v_2 \wedge v_3 = a \,v_1 \cdot v_2 \wedge v_3 + 0 + 0 $$ from which you can find $a$. Similarly $b$ and $c$.