Question

1. Consider three vectors u=(-1,5) v=(2.7,3.8) w=(4.2,-6). find the coordinates of vectors
a)u+v
b)3u-2v-5w
c)u+v+w​

Answer 1

a) The coordinates of the combination of vectors are $\vec u + \vec v = (1.7, 8.8)$

b) The coordinates of the combination of vectors are $3\vec u -2\vec v - 5\vec w = (-25.2, 37.8)$

c) The coordinates of the combination of vectors are $\vec u + \vec v + \vec w = (5.9, 2.8)$

The sum of vectors consists in sum the values corresponding to each component, that is:

$\vec u = (u_{x}, u_{y})$, $\vec v = (v_{x}, v_{y})$

$\vec r = (u_{x} + v_{x}, u_{y}+v_{y})$ (1)

And the multiplication of a vector by scalar is described below:

$\vec v = (v_{x}, v_{y})$, $\alpha \in \mathbb{R}$

$\vec r = \alpha\cdot \vec v$

$\vec r = \alpha \cdot (r_{x}, r_{y})$

$\vec r = (\alpha\cdot v_{x}, \alpha\cdot v_{y})$ (2)

If we know that $\vec u = (-1, 5)$, $\vec v = (2.7, 3.8)$ and $\vec w = (4.2,-6)$, then the coordinates of vectors are:

a) $\vec u + \vec v$

$\vec r = (-1, 5) +(2.7, 3.8)$

$\vec r = (1.7, 8.8)$

b) $3\vec u -2\vec v - 5\vec w$

$\vec r = 3\cdot(-1,5) -2\cdot (2.7, 3.8) -5\cdot (4.2, -6)$

$\vec r = (-3, 15) - (5.4, 7.2) - (16.8, -30)$

$\vec r = (-25.2, 37.8)$

c) $\vec u + \vec v + \vec w$

$\vec r = (-1,5) + (2.7, 3.8) +(4.2, -6)$

$\vec r = (5.9, 2.8)$

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