Question Video: Calculating the Scalar Product of Two Vectors Given in Component Form Physics

A constant force of 𝐅 = (1𝐒 + 4𝐣) N acts on an object, causing it to move. After an amount of time, the displacement of the object from its initial position is 𝐝 = (5𝐒 + 2𝐣) m. Calculate 𝐅 β‹… 𝐝.

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Video Transcript

A constant force of 𝐅 equals one 𝐒 plus four 𝐣 newtons acts on an object, causing it to move. After an amount of time, the displacement of the object from its initial position is 𝐝 equals five 𝐒 plus two 𝐣 meters. Calculate 𝐅 dot 𝐝.

So we have some object, and we have a constant force acting on that object that results in a displacement. The force and displacement vectors have been specified for us in component form. Component form means that for two-dimensional vectors, they’re specified by their π‘₯- and their 𝑦-components. For a general vector 𝐀, we can write it as the magnitude of its π‘₯-component, 𝐴 subscript π‘₯, multiplied by the unit vector in the π‘₯-direction, that’s 𝐒, plus the magnitude of its 𝑦-component, 𝐴 subscript 𝑦, multiplied by the unit vector in the 𝑦-direction, that’s 𝐣.

If we do the same for another general vector 𝐁, then we can define the scalar product of these two vectors as follows. The scalar product of two vectors is equal to the product of the π‘₯-components of those vectors plus the product of their 𝑦-components. For our force vector 𝐅, the π‘₯-component is one newton and the 𝑦-component is four newtons. For our displacement vector 𝐝, the π‘₯-component is five meters and the 𝑦-component is two meters. We can draw these vectors as arrows to get a sense of what’s going on.

Let’s imagine we have some object represented by this little blue blob here. The force vector has one unit in the π‘₯-direction and four units in the 𝑦-direction. The displacement vector has five units in the π‘₯-direction and two in the 𝑦-direction. Note that since force and displacement have different units, newtons and meters, respectively, these two arrows aren’t actually on the same scale as each other. This diagram just enables us to visualize their relative directions.

Right, let’s get back to the question and calculate the scalar product 𝐅 dot 𝐝. We have the product of the π‘₯-components, so that’s one and five, plus the product of the 𝑦-components, so that’s four and two. So we have that 𝐅 dot 𝐝 is equal to one multiplied by five plus four multiplied by two.

We also need to take care of the units here. Since force has units of newtons and displacement has units of meters, then the scalar product 𝐅 dot 𝐝 must have units of newton meters. If we wanted, we could equivalently write this as joules, the unit of energy.

Doing the maths, the product of the π‘₯-components is five and the product of the 𝑦-components is eight. The sum of five and eight gives us 13. And so we have our answer to the question that the scalar product 𝐅 dot 𝐝 is equal to 13 newton meters, or equivalently 13 joules.

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