Name: Introduction to Vectors
Uploaded: 2019-07-19
Duration: 6 min 59 s
Description: Most of you might have seen vectors in Physics by now. Here, the same concept is explained through the lens of a Mathematician. Watch this video to learn more!!

Question

Show that the following vectors are coplanar:

$\vec{a} - 2 \vec{b} + 3 \vec{c}, 2 \vec{a} + 3 \vec{b} - 4 \vec{c}, - 7 \vec{b} + 10 \vec{c}$

Accepted Answer

Vectors $\vec{A B}, \vec{B C}$ and $\vec{C A}$ are coplanar if $\exists x, y, z \in ℝ$ such that $x \vec{A B} + y \vec{B C} + z \vec{C A} = 0$ where $A, B, C$ are non-collinear.

We have to check whether the vectors $\vec{a} - 2 \vec{b} + 3 \vec{c}, 2 \vec{a} + 3 \vec{b} - 4 \vec{c}$ and $- 7 \vec{b} + 10 \vec{c}$ are coplanar.

We assume that $\vec{a}, \vec{b}, \vec{c}$ are not the collinear vectors.

Let $x (\vec{a} - 2 \vec{b} + 3 \vec{c}) + y (2 \vec{a} + 3 \vec{b} - 4 \vec{c}) + z (- 7 \vec{b} + 10 \vec{c}) = 0$

$\Rightarrow x \vec{a} - 2 x \vec{b} + 3 x \vec{c} + 2 y \vec{a} + 3 y \vec{b} - 4 y \vec{c} - 7 z \vec{b} + 10 z \vec{c} = 0$

$\Rightarrow (x + 2 y) \vec{a} + (- 2 x + 3 y - 7 z) \vec{b} + (3 x - 4 y + 10 z) \vec{c} = 0$

$\Rightarrow x + 2 y = 0 . . . (1)$ , $- 2 x + 3 y - 7 z = 0 . . . (2)$ and $3 x - 4 y + 10 z = 0 . . . (3)$

Solving $(1), (2)$ and $(3)$ simultaneously, we get $y = - \frac{x}{2}$ and $z = - \frac{x}{2}$ .

So for $x \neq 0$ , we can find $y, z \neq 0$ such that $x (\vec{a} - 2 \vec{b} + 3 \vec{c}) + y (2 \vec{a} + 3 \vec{b} - 4 \vec{c}) + z (- 7 \vec{b} + 10 \vec{c}) = 0$ .

Hence, $\vec{a} - 2 \vec{b} + 3 \vec{c}, 2 \vec{a} + 3 \vec{b} - 4 \vec{c}, - 7 \vec{b} + 10 \vec{c}$ are coplanar vectors.

Show that the following vectors are coplanar:
$\vec{a} - 2 \vec{b} + 3 \vec{c}, 2 \vec{a} + 3 \vec{b} - 4 \vec{c}, - 7 \vec{b} + 10 \vec{c}$

Important Questions on Vectors

Learn and Prepare for any exam you want !

Show that the following vectors are coplanar: a→-2b→+3c→, 2a→+3b→-4c→, -7b→+10c→

Important Questions on Vectors

Learn and Prepare for any exam you want !

Show that the following vectors are coplanar:
$\vec{a} - 2 \vec{b} + 3 \vec{c}, 2 \vec{a} + 3 \vec{b} - 4 \vec{c}, - 7 \vec{b} + 10 \vec{c}$