# Populate item in works with

Populate item of 2 vectors on aircraft can be specified as item of sizes. of those vectors and also cosine of angle in between them.

In cartesian works with dot item of vectors with works with $(x_1, y_1)$ and also. $(x_2, y_2)$ amounts to $x_1x_2 + y_1y_2$.

Just how to confirm it?

The dot item is stable under turnings, we might consequently revolve our coordinate system to make sure that v is along the x-axis. In this instance, $v = (|v|, 0)$. Allowing $w = (x,y)$ we have (making use of the definition of dot item in Cartesian works with) $v \cdot w = |v| x$. Yet what is $x$? Well, if you attract the image and also allow $\theta$ be the angle in between v and also w, after that we see that $\cos \theta = x/|w||$ to make sure that $x = |w| \cos \theta$. Hence $v\cdot w = |v||w| \cos \theta$.

I intend you intend to confirm that 2 your interpretations of dot item coincide. We start with definition of dot item as $(\vec{u}, \vec{v}) = |\vec{u}| |\vec{v}| \cos \theta$. We start with definition of dot item as $(\vec{u}, \vec{v}) = |\vec{u}| |\vec{v}| \cos \theta$ and also confirm that it additionally pleases $(\vec{u}, \vec{v}) = x_1 x_2 + y_1 y_2$.

In the beginning you can confirm that dot item is straight : $(\vec{v_1}, \vec{v_2} + \alpha \vec{v_3}) = (\vec{v_1}, \vec{v_2}) + \alpha (\vec{v_1}, \vec{v_3})$. This holds true due to the fact that $(\vec{v_1}, \vec{v_2})$ amounts to the item of $|\vec{v_1}|$ and also estimate of $\vec{v_2}$ on $\vec{v_1}$. Estimate of amount of vectors amounts to amount of estimates. Therefore dot item is straight.

Allow $\vec{e_1}$ and also $\vec{e_2}$ be vectors with works with $(1, 0)$ and also $(0, 1)$.

Afterwards if $\vec{v_1} = x_1 \vec{e_1} + y_1 \vec{e_2}$ and also $\vec{v_2} = x_2 \vec{e_2} + y_2 \vec{e_2}$ after that by linearity of dot item we have $(\vec{v_1}, \vec{v_2}) = x_1 x_2 (\vec{e_1}, \vec{e_1}) + x_1 y_2 (\vec{e_1}, \vec{e_2}) + x_2 y_1 (\vec{e_2}, \vec{e_1}) + x_2 y_2 (\vec{e_2}, \vec{e_2})$. Given that $(\vec{e_1}, \vec{e_1}) = (\vec{e_2}, \vec{e_2}) = 1$ and also $(\vec{e_1}, \vec{e_2}) = (\vec{e_2}, \vec{e_1}) = 0$ we have $(\vec{v_1}, \vec{v_2}) = x_1 x_2 + y_1 y_2$.

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