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:

$4 \vec{a} + 5 \vec{b} + \vec{c}, 5 \vec{a} + 9 \vec{b} + 4 \vec{c}, - \vec{b} - \vec{c}$

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 $4 \vec{a} + 5 \vec{b} + \vec{c}, 5 \vec{a} + 9 \vec{b} + 4 \vec{c}$ and $- \vec{b} - \vec{c}$ are coplanar.

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

Let $x (4 \vec{a} + 5 \vec{b} + \vec{c}) + y (5 \vec{a} + 9 \vec{b} + 4 \vec{c}) + z (- \vec{b} - \vec{c}) = 0$

$\Rightarrow 4 x \vec{a} + 5 x \vec{b} + x \vec{c} + 5 y \vec{a} + 9 y \vec{b} + 4 y \vec{c} - z \vec{b} - z \vec{c} = 0$

$\Rightarrow (4 x + 5 y) \vec{a} + (5 x + 9 y - z) \vec{b} + (x + 4 y - z) \vec{c} = 0$

$\Rightarrow 4 x + 5 y = 0 . . . (1)$ , $5 x + 9 y - z = 0 . . . (2)$ and $x + 4 y - z = 0 . . . (3)$

Solving $(1), (2)$ and $(3)$ simultaneously, we get $y = \frac{- 4 x}{5}$ and $z = \frac{- 11 x}{5}$ .

So for $x \neq 0$ , we can find $y, z \neq 0$ such that $x (4 \vec{a} + 5 \vec{b} + \vec{c}) + y (5 \vec{a} + 9 \vec{b} + 4 \vec{c}) + z (- \vec{b} - \vec{c}) = 0$ .

Hence, $4 \vec{a} + 5 \vec{b} + \vec{c}, 5 \vec{a} + 9 \vec{b} + 4 \vec{c}$ and $- \vec{b} - \vec{c}$ are coplanar.

Show that the following vectors are coplanar:
$4 \vec{a} + 5 \vec{b} + \vec{c}, 5 \vec{a} + 9 \vec{b} + 4 \vec{c}, - \vec{b} - \vec{c}$

Important Questions on Vectors

Learn and Prepare for any exam you want !

Show that the following vectors are coplanar: 4a→+5b→+c→, 5a→+9b→+4c→, -b→-c→

Important Questions on Vectors

Learn and Prepare for any exam you want !

Show that the following vectors are coplanar:
$4 \vec{a} + 5 \vec{b} + \vec{c}, 5 \vec{a} + 9 \vec{b} + 4 \vec{c}, - \vec{b} - \vec{c}$