Vector addition and Pythagorean theorem

Question

Finding length or magnitude using vector addition and the Pythagorean theorem.

I am trying to understand why vector addition and the Pythagorean theorem are giving different results?

Vector Addition :

According to diagram (A) : $\vec{a} + \vec{b} = \vec{c}$

now suppose : magnitude of $\vec{a} = 3$, magnitude of $\vec{b} = 4$ then

$\|\vec{c}\| = \|\vec{a}\| + \|\vec{b}\|$

$\|\vec{c}\| = 3 + 4$

magnitude of c = 7

Pythagorean Theorem

Now when we consider this as a triangle shown in diagram (B)

Similarly, length of $a = 3$, length of $b = 4$

so according to Pythagorean theorem

$\text{hypotenuse}^2 = \text{opposite}^2 + \text{adjacent}^2$

$c^2 = a^2 + b^2$

i.e.

$c^2 = 3^2 + 4^2$

$c = \sqrt{9 + 16} = \sqrt{25}$

length of c = 5

Why there is inconsistency in the results? I am doing something wrong?

Thanks!

Answer 1

Vector Magnitude

The magnitude of a vector $(\vec{a} + \vec{b})$ is obtained in a similar way to that for Pythagorean triangles. It is incorrect to simply add up magnitudes of $\vec{a}$ and $\vec{b}$.

Rather, $\| \vec{c} \| = \sqrt {\| \vec{a} \| + \| \vec{b} \|} $ (See link).

Khan Academy

As @JMoravitz has already demonstrated.