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Jetzt kostenlos anmeldenThere are two types of projections that you need to be aware of at this stage: scalar projections and vector projections.
Scalar projection simply gives the length in a particular direction. The result of this is a scalar that quantifies this amount. On the other hand, vector projection 'projects' the length of one vector in the direction of another. One way to think of it is as if the 'shadow' of a vector is cast over the top of another vector.
We will start with the simplest to understand conceptually. The scalar projection of a vector finds how much, as a scalar, of the vector is in a particular direction. It is found using the dot product of the vector with the unit vector in the direction in question.
The scalar projection of a vector onto unit vector is the scalar given by the dot product: or
Find the scalar projection of vector in the horizontal direction.
Solution
The 'horizontal direction' is along the x-axis, therefore the unit vector we will be using is . Vector in vector notation is .
Intuitively, you may have already realized that the scalar projection must be 3 since by definition is comprised of 3 units in the horizontal direction (and 4 vertically).
We can show this using the dot product too:
Therefore, the scalar projection of in the horizontal direction is equal to 3.
A vector projection is the projection of one vector onto another. It takes the length of one vector and projects it in the direction of another, creating a new vector with the direction of the second.
Vector b is known as the projection of a in the x-axis direction (figure 2). If you look at figure 3, vector c is a vector in the x-axis direction, so b is also the projection of a in the direction of vector c.
In mathematical notation, this is written as . We know that the vector must be equal to , therefore is such that is orthogonal to vector c.
This orthogonality is a property that is essential to finding the projection of a onto L. Remember the 'dot product'? Since is orthogonal to line L, the 'dot product' of the two must be equal to zero. Using this information, we can derive a formula for the vector projection of a in the direction of b.
To understand fully what the projection vector does, it may be useful to see how to derive it. First, let's consider a vector v lying on line L. Since is in the direction of L, we can write it as a scalar multiple of vector v.
v is some vector in the 2D space
c is some constant
Using the dot product:
Since the dot product of a vector with itself is equal to the length of that vector squared, we get:
.
And so, .
The vector projection of vector a onto vector v is .
We have vector and vector . Find the projection of x onto v.
Using the formula:
Therefore, .
is represented by the blue arrow in the diagram below.
Scalar projection gives the scalar length in a particular direction. On the other hand, vector projection 'projects' the length of one vector in the direction of another.
Two examples of projection are scalar projection and vector projection.
Scalar projection is given by the dot product.
Both scalar projection and vector projection are done by applying their respective formula.
Projections in geometry do not fall under the scope of the article. Nevertheless, they are spherical, cylindrical, conical, and azimuthal projections.
What are the two types of projections?
Scalar projection and vector projection.
Explain vector projection in words.
Vector projection 'projects' the length of one vector in the direction of another.
Explain scalar projection in words.
The scalar projection of a vector is how much the vector is in a particular direction.
(Revision) What is the difference between a scalar and a vector?
A scalar is a quantity with only magnitude (e.g. length). A vector has both magnitude and direction.
Define orthogonality.
If two vectors are orthogonal, they are perpendicular to each other (i.e.they are at right angles).
What is the dot product of orthogonal vectors?
Zero
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