How To Add Three Vectors Using Components

Now that we have learned to pause a vector into components, we tin can begin calculation vectors using components.

The wonderful affair about vector components is that once we chose a coordinate system, all x-components of vectors point in the same direction. This means that we can add the x-components of two vectors by just adding them. The same holds true for the y-components.

In order to add two random vectors, nosotros simply interruption each into components. We so add the x-components together. And then we add together the y-components together. Finally, we use the Pythagorean theorem to find the resultant and the trig functions to find the direction.

You can review finding components and adding perpendicular vectors.

This blithe .gif outlines the process.

Animated .gif of adding vectors using components.

Adding Vectors Using Components

We commencement break the two vectors, \(\colour{black}{\vec{A}}\) and \(\color{black}{\vec{B}}\) into components.

\(\color{black}{\vec{A} = \vec{A_x} + \vec{A_y}}\) and \(\color{black}{\vec{B} = \vec{B_x} + \vec{B_y}}\)

It might assist to write these equally

\(\color{black}{\vec{A} = A_x \lid{x} + A_y \hat{y}}\) and \(\color{black}{\vec{B} = B_x \hat{x} + B_y \hat{y}}\).

The 10-component of the resulting vector \(\color{black}{\vec{C}}\) is simply the sum of the x-components of \(\colour{black}{\vec{A}}\) and \(\color{black}{\vec{B}}\)

\(\color{black}{\vec{C_x} = A_x \lid{x} + B_x\hat{x}}\).

Likewise, the y- component is given by

\(\color{black}{\vec{C_y} = A_y \chapeau{y} + B_y\lid{y}}\).

We could write this equally

\(\color{black}{\vec{C} = (A_x + B_x) \hat{ten} + (A_y + B_y) \hat{y}}\).

Because the 10-components are parallel, we could fifty-fifty go so far every bit to just write

\(\colour{black}{C_x = A_x + B_x}\) and \(\color{black}{C_y = A_y + B_y}\).

In one case nosotros take the values of \(\color{black}{C_x}\) and \(\color{blackness}{C_y}\), we use the Pythagorean theorem to find C…

\(\color{black}{C^2 = C^2_x + C^2_y}\).

Finally, we utilise the tangent function to discover \(\color{black}{\theta_c}\)…

\(\color{black}{\tan \theta = \frac{C_y}{C_x}}\).