8.4
The Dot Product
The dot product of two vectors is a scalar. It is calculated by multiplying their magnitudes—shown with double vertical bars and found using the square root of the sum of squared components—and the cosine of the angle between them.
Consider a box resting on a ramp and being pulled by a rope at an angle. The pulling force has a magnitude and direction, forming a vector.
The ramp defines a direction, represented by a unit vector along the inclined direction.
To determine the effective pulling force along the ramp, the dot product is used. It equals the force magnitude times the unit vector’s magnitude, and the cosine of the angle. Since the unit vector’s magnitude is one, this simplifies to the force magnitude times the cosine of the angle.
This value also represents the component of the rope’s force in the ramp’s direction.
When the pull aligns with the ramp, the entire force is effective because the cosine of a zero angle is one.
In this case, the force matches its component along the unit vector, known as the projection vector. This vector represents how much of the pulling force acts in that direction.
Measuring how one directional quantity affects another along a specific path involves comparing their orientation and strength. When two such quantities are represented using direction and amount, a numerical result is computed to show how much one acts along the path of the other. This result comes from a rule combining both inputs' horizontal and vertical parts and adding the results.
This calculation gives a single value that grows larger when both inputs point in similar directions and becomes zero when they are at right angles. It gives a complete measure of how aligned the two inputs are.
In another form, this result is also found by multiplying the total sizes of each input with a value that measures how closely they point in the same direction.
This version helps when the sizes and turning between inputs are known. The closer the directions are to each other, the larger the outcome, reaching its highest value when they point the same way. These expressions provide clear ways to compute how strongly one input acts along the path defined by another.
The dot product of two vectors is a scalar. It is calculated by multiplying their magnitudes—shown with double vertical bars and found using the square root of the sum of squared components—and the cosine of the angle between them.
Consider a box resting on a ramp and being pulled by a rope at an angle. The pulling force has a magnitude and direction, forming a vector.
The ramp defines a direction, represented by a unit vector along the inclined direction.
To determine the effective pulling force along the ramp, the dot product is used. It equals the force magnitude times the unit vector’s magnitude, and the cosine of the angle. Since the unit vector’s magnitude is one, this simplifies to the force magnitude times the cosine of the angle.
This value also represents the component of the rope’s force in the ramp’s direction.
When the pull aligns with the ramp, the entire force is effective because the cosine of a zero angle is one.
In this case, the force matches its component along the unit vector, known as the projection vector. This vector represents how much of the pulling force acts in that direction.
From Chapter 8:
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8.4 : The Dot Product
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