Name: Parallelogram Law for Addition of Two Vectors
Uploaded: 2019-07-19
Duration: 5 min 1 s
Description: Now, we know one of the methods to add the two vectors. It is time to learn another law to add the two vectors. Explore this video to learn about it.

Question

Prove by vector method that the altitude of a triangle are concurrent.

Accepted Answer

Let $A B C$ be a triangle and $A D, B E, C F$ are altitudes intersecting at $O$

In order to prove that the altitudes are concurrent it is sufficient to prove that $O C$ is perpendicular to $A B$ .

Taking $O$ as the origin, let the position vectors of $A, B, C$ be $\vec{a}, \vec{b}, \vec{c}$ respectively.

So, $\vec{O A} = \vec{a}, \vec{O B} = \vec{b}, \vec{O C} = \vec{c}$

Now, $\vec{A D} ⊥ \vec{B C}$

$\Rightarrow \vec{O A} ⊥ \vec{B C}$

$\Rightarrow \vec{O A} \cdot \vec{B C} = 0$

$\Rightarrow (\vec{a}) \cdot (\vec{b} - \vec{c}) = 0$

$\Rightarrow \vec{a} \cdot \vec{b} - \vec{a} \cdot \vec{c} = 0 . . . . (1)$

And $\vec{B E} ⊥ \vec{C A}$

$\Rightarrow \vec{O B} ⊥ \vec{C A}$

$\Rightarrow \vec{O B} \cdot \vec{C A} = 0$

$\Rightarrow \vec{b} \cdot (\vec{a} - \vec{c}) = 0$

$\Rightarrow \vec{b} \cdot \vec{a} - \vec{b} \cdot \vec{c} = 0 . . . . (2)$

Adding $(1)$ and $(2)$ , we get

$\vec{a} \cdot \vec{c} - \vec{b} \cdot \vec{c} = 0 \Rightarrow (\vec{a} - \vec{b}) \cdot \vec{c} = 0$

$\Rightarrow \vec{B A} \cdot \vec{O C} = 0$

$\Rightarrow \vec{O C} ⊥ \vec{B A}$

Therefore, the three altitudes are concurrent.

Prove by vector method that the altitude of a triangle are concurrent.

Important Questions on Vectors

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