A Visual Representation of the Vector Scalar Product?

[avocadogirl]

To any teachers or students, either instructing or taking, a Calculus-based Physics I course:

I tutor a calculus-based general physics course in kinematics, and similar topics, and, I recently had a student approach me about his inability to grasp the scalar/dot product, in vector operations. His concerns seemed to convey that he was having trouble "visualizing" a dot product, comprehending real-to-life situations where it would be applicable and, understanding why the formula relating the dot product of two vectors always involved the cosine of the angle between them.

I've heard it explained before, in loose terms, that one could look at the result of a scalar/dot product as an area, and the result of the vector/cross product as a volume. I'm hesitant to endorse that representation because, I fail to see the accuracy of the comparison.

So, if any of you viewers can offer, what you feel to be, an effective example problem or explanation, from either an experience teaching or enduring the presentation of similar subject matter, it would be greatly appreciated. Thank You All!

 

[tiny-tim]

Welcome to PF!

… he was having trouble "visualizing" a dot product, comprehending real-to-life situations where it would be applicable and, understanding why the formula relating the dot product of two vectors always involved the cosine of the angle between them.

I've heard it explained before, in loose terms, that one could look at the result of a scalar/dot product as an area, and the result of the vector/cross product as a volume. I'm hesitant to endorse that represe

Hi avocadogirl! Welcome to PF!

No, I think cross product is an area, and the triple (scalar) product is a volume …

I'd go with dot prouct being half the "Pythagorean excess" of a triangle, c² - (a² + b²)

 

[Tac-Tics]

It's not area, as tiny-tim said, but try this.

If e and v are both vectors, e * v is the length of the vector projected orthogonally onto e times the length of e. (Or the length of the projected vector in UNITS of e).

OK, that is a bit confusing how I worded it, but start off with explaining how projection works. Draw the "arrows" of the two vectors e and v at the origin. Then, draw a line segment starting at the pointed-end of v to the line that lies along e, such that the line segment and e are perpendicular. Extend or contract e to it exactly meets the line segment you just drew. This "extended version" of e is the projection of v onto e.

This is easy to explain with a diagram. It's something like "what would the shadow of v look like?"

Once you have explained projection, the dot product is just "the length of the projection of v in units of e". If e is a unit vector, then the dot product is just the length of the projection.

The two most important properties of the dot product are its relation to cosine, which you have mentioned already to your student, but also that it embodies "orthogonality." If you draw e and v, and e and v happen to be perpendicular to each other, then the projection becomes a single point, and the length of the projection is 0. Orthogonal vectors are super important and it's nice to be able to recognize them immediately.

 

[avocadogirl]

Thank you, both! I appreciate the warm welcome to Physics Forum, which I think is an amazing academic resource, and, I found both replies to be exceptionally helpful.

Thanks again!
-avocadogirl

 

[qntty]

For the area definition of a cross product, $|\vec{a} \times \vec{b}|$ is the area of the parallelogram with side lengths $|\vec{a}|$ and $|\vec{b}|$ (the same parallelogram that you use to add the vectors)

 

[avocadogirl]

Thank you. I'll include that explanation of the cross product in my reply to this young man.

So, I assume that, since the resulting vector of the cross product is perpendicular to both of the vectors being multiplied, the resulting vector is a vector which represents the area of lengths |a| and |b|?