The Dot Product of two products in $\R^2$

<aside> 💡 For two general vectors $(x_1,y_1)$ and $(x_2,y_2)$ in $\R^2$, the dot product of these two vectors is a scalar computed as follows: $(x_1,y_1).(x_2,y_2)=x_1x_2+y_1y_2$

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The Length of a Vector in $\R^2$

<aside> 💡 Length of a vector $v(x,y)$ is given by $d=\sqrt{x^2+y^2}$

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The Relation between Length and Dot product in $\R^2$

<aside> 💡 The length of a vector is the square root of the dot product of the vector with itself.

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The angle between two vectors in $\R^2$

• The angle between the vectors $u$ and $v$ measures how far the direction is of $v$ from $u$(or vice versa).

The Dot product and the angle between two vectors in $\R^2$

Let $u$ and $v$ be two vectors in $\R^2$. Then we can compute the angle $\theta$ between the vectors $u$ and $v$ using the dot products as:

The angle between two vectors in $\R^3$ and the dot product

The angle between the vectors $u$ and $v$ in $\R^3$ is the angle between them computed by passing a plane through them.