A Visual Representation of the Vector Scalar Product?

In summary, a student was struggling to understand the scalar/dot product in vector operations, specifically in visualizing its application in real-life situations and understanding the formula involving the cosine of the angle between two vectors. Some explanations compare the dot product to an area, but this may not be accurate. Another helpful way to understand the dot product is as the length of the projection of one vector onto another. The cross product, on the other hand, can be thought of as the area of a parallelogram with side lengths determined by the two vectors being multiplied.
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!

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

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)

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.

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!

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)

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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|?

1. What is a Vector Scalar Product?

A Vector Scalar Product is a mathematical operation that combines a vector and a scalar (a single value) to produce a new vector. This new vector has the same direction as the original vector, but its length is multiplied by the scalar value.

2. How is a Vector Scalar Product represented visually?

A Vector Scalar Product is often represented visually using a diagram or graph. The original vector is shown as an arrow, with its direction and magnitude indicated. A scalar value is then represented as a number, often written next to the arrow. The resulting vector is then shown as a new arrow, with its length multiplied by the scalar value.

3. What is the formula for calculating a Vector Scalar Product?

The formula for calculating a Vector Scalar Product is: c * v = cv, where c is the scalar value and v is the vector. This means that each component of the vector is multiplied by the scalar value to produce the new vector. For example, if v = (2, 3) and c = 4, then the resulting vector would be cv = (8, 12).

4. What is the significance of the Vector Scalar Product?

The Vector Scalar Product is significant in mathematics and physics because it allows us to scale vectors, or change their magnitude, while still maintaining their direction. This is useful in many real-world applications, such as calculating forces, velocities, and displacements in physics problems.

5. What are some common uses of the Vector Scalar Product?

Some common uses of the Vector Scalar Product include calculating work, power, and torque in physics, as well as determining the direction and magnitude of forces acting on an object. It is also used in computer graphics to scale and rotate objects in three-dimensional space.

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