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16.2: Vector Addition

  • Page ID
    55332
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    The most common thing we will need to do with many vector quantities is to add them up. The sum of these vector quantities is the net vector quantity. For example, if we have a number of forces acting on a body, the sum of those forces is known as the net force.

    The sum of any number of vectors can be determined geometrically using the following strategy. Starting with one of the vectors as the base, we redraw the second vector so that the tail of the second vector begins at the tip of the first vector. We can repeat this with a third vector, a forth vector and so on, putting the tail of each vector at the tip of the last vector until we have added taken all vectors into account. Once the vectors are all drawn tip to tail, the sum of all the vectors will be the vector connecting the tail of the first vector to the tip of the last vector.

    The first quadrant of a two-dimensional Cartesian coordinate system. A vector F_1 extends to the right and sharply upwards from the origin. A second vector F_2 is placed with its tail at the head of F_1, and extends further to the right and upwards is placed. The vector F_net that is the sum of these two vectors extends from the origin to the head of vector F_2.
    Figure \(\PageIndex{1}\): The geometric addition of vectors involves putting the vectors tip to tail as shown above.

    In practice, the easiest way to determine the magnitude and direction of the sum of the vectors is to add the vectors in component form. This starts by separating each vector into \(x\), \(y\), and possibly \(z\) components. As we can see in the diagram below, the \(x\) component of the sum of all the vectors will be the sum of all the \(x\) components of the individual vectors. Similarly, the \(y\) and \(z\) components of the sum of the vectors will be the sum of all the \(y\) components and the sum of all the \(z\) components respectively.

    The set of vectors from Figure 1 above is redrawn to include the x- and y-components of each vector. The x-component of F_net equals F_1x + F_2x, and the y-component of F_net equals F_1y + F_2y.
    Figure \(\PageIndex{2}\): By summing all the components in a given direction, we can find the component of the sum of the vectors in that direction.

    Once we find the sum of the components in each direction, we can either leave the net vector in component form, or we can use the Pythagorean theorem and inverse tangent functions to convert the vector back into a magnitude and direction as detailed on the previous page on vectors.

    Figure \(PageIndex{1}\): Vide lecture covering this section, delivered by Dr. Jacob Moore. YouTube source: https://youtu.be/0tv92MX2_ro.

    Example \(\PageIndex{1}\)

    Determine the sum of the force vectors in the diagram below. Leave the sum in component form.

    Three two-dimensional vectors radiate out from a single point. One vector, with magnitude 5 kN, points directly to the right. A second vector, with magnitude 3 kN, points up and to the right at 45° above the horizontal. The third vector, with magnitude 6 kN, points down and to the left at 30° clockwise from the vertical.
    Figure \(\PageIndex{3}\): problem diagram for Example \(\PageIndex{1}\). Three two-dimensional vectors radiate out from a single point.
    Solution:
    Video \(\PageIndex{2}\): Worked solution to example problem \(\PageIndex{1}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/FwC8ntactEQ.

    Example \(\PageIndex{2}\)

    Determine the sum of the force vectors in the diagram below. Give the sum in terms of a magnitude and a direction.

    Three two-dimensional vectors radiate out from a single point. One vector, with magnitude 5 kN, points directly towards the bottom of the page. A second vector, with magnitude 6 kN, points down and to the left at 30° clockwise from the vertical. The third vector, with magnitude 5 kN, is 90° clockwise from that second vector.
    Figure \(\PageIndex{4}\): problem diagram for Example \(\PageIndex{2}\). Three two-dimensional vectors radiate out from a single point.
    Solution:
    Video \(\PageIndex{3}\): Worked solution to example problem \(\PageIndex{2}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/Jj8mCV7rdas.

    Example \(\PageIndex{3}\)

    Determine the sum of the force vectors in the diagram below. Leave the sum in component form.

    Two vectors radiate out from the origin of a three-dimensional Cartesian coordinate system, with the x- and y-axes lying in the plane of the screen and the z-axis extending out of the screen. The vector F_1, with magnitude 600 lbs, points 30° out of the xy-plane towards the viewer and then points upwards and rightwards at 45° above the xz-plane. The vector F_2, with magnitude 300 lbs, is directed out of the screen towards the viewer at 40° above the z-axis.
    Figure \(\PageIndex{5}\): problem diagram for Example \(\PageIndex{3}\). Two vectors radiate out from the origin of a three-dimensional Cartesian coordinate system.
    Solution:
    Video \(\PageIndex{4}\): Worked solution to example problem \(\PageIndex{3}\), provided by Dr. Jacob Moore. YouTube source: https://youtu.be/PZzx3eQp6iQ.

    This page titled 16.2: Vector Addition is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Jacob Moore & Contributors (Mechanics Map) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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