5/31/2023 0 Comments Multiply a scalar with vector codeIn standard form here, x component one, two, three, and then y component two, three, four, five and six. ![]() And so we see the resulting vector, we could call this vector three w, it's gonna have an x component of three and a y component of six. Three times one, and then three times two, and so this is going to be equal to, this is going to be equal to, three times one is three, three times two is six. ![]() So this is going to be equal to, we have a one and a two,Īnd we're gonna multiply each of those times the three. Multiply each of the components times that scalar. Now, the convention we useįor multiplying a scalar times a vector is, you just So, for example, we could think about, what is three times w going to be? Three times w. When you were four years old, those are scalars. You could think of just the numbers that you started learning What do we mean by a scalar? Well, a vector is something that has a magnitude and a direction. Otherwise, it's nice to just put its initial Have the same vector and I could shift itĪround as long as I have the same length of the arrow and it's pointing in the same direction. To be right over there, the vector, in standard, graphing it in standard form or visualizing it in standardįorm, would look like that. This vector is going to look like, its initial point is right here, its terminal point is going So its x coordinate is one, its y coordinate is going to be two. And then its terminal point would be at the point one comma two. And so, if I were to draw this vector in standard form, I would Let's say its x component is one and its y component Let's say I have the vector w, and let me give it an x component. Now, what am I talking about when I say, multiplying a scalar times a vector? Well, let me set up a little This has a simple (though not entirely useful, at least not in physics) geometric interpretation in terms of the parallelogram defined by the two vectors:įigure 1.2.Wanna do in this video is give ourselves some practice,Īnd hopefully some intuition, on multiplying a scalar times a vector. Because this is distinct from the scalar product, we use a different mathematical notation as well – a cross rather than a dot (giving it an alternative name of cross product). For reasons that will be clear soon, this type of product is referred to as a vector product. If we take a product like before, multiplying this perpendicular piece by the magnitude of the other vector, we get an expression similar to what we got for the scalar product, this time with a sine function rather than a cosine. ![]() If the scalar product involves the amount of one vector that is parallel to the other vector, then it should not be surprising that our other product involves the amount of a vector that is perpendicular to the other vector.įigure 1.2.2 – Portion of One Vector Perpendicular to Another Finally, with the magnitudes of the vectors and the angle between the vectors, we could finally plug into our scalar product equation.Īs mentioned earlier, there are actually two ways to define products of vectors. Then we would need to compute the magnitudes of the two vectors. ![]() Then from those two angles, we need to figure out the angles between the two vectors. If we didn’t have this simple result, think about what we would have to do: We would need to calculate the angles each vector makes with (say) the \(x\)-axis.
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