Topic 4: Vectors
What can be said about line when they are coincident?
The lines share all their points, thus they are the same line, even if they are expressed differently
A vector can be written in column form, how is this done?
This vector is: x units in the x-direction y units in the y-direction z units in the z direction
What is a zero vector?
* The origin v+(-v)=0=(-v)+v
How can a position vector be written in terms of two other vectors?
- Switching the letters is the same as using minus
How are vectors subtracted graphically?
- They are 180° degrees
What type of angle should always be given, when calculating the angle between two vectors?
Acute angle. If answer is obtuse then θ=180°-obtuse angle
Is the vector product commutative?
The vector product of two vectors is not commutative i.e. u×v=v×u
When are two vectors equal?
Two vectors are equal if and only if all their components are equal. - The same direction - The same magnitude
What are the properties of the vector product?
1. The vector product of two vectors is itself a vector, this is has a magnitude and a direction. 2. The vector product of two vectors is not commutative, i.e. u×v ≠ v×u 3. In fact, u×v = -v×u. That is, they are parallel but point in the opposite directions 4. The vector product of two vectors u×v is oriented perpendicular to the plane containing u and v
Define a position vector
A position vector is a vector that starts at the origin and end at a designated point → The components of a position vector are the coordinates of the point they are associated with.
Is addition of vectors commutative?
Addition of vectors is commutative: v+w = w+v
How can you find the minimum distance between two objects going at a constant speed?
Find the minimum point of the graph, for the magnitude/distance, d
How are vectors added graphically?
Geometrically we place the "tail" of one to the "head" of the other and the resultant vector from the first vector's tail to the last vector's head is the sum of these two vector.
How do you show that two vectors are parallel?
If two vectors are parallel, then a scalar exists such that v=kw, which means that for all the vectors lines, there will be a common value for k k<0: The vectors point in opposite directions k>0: The vectors point in the same direction Also: If v‖w then v×w = |v||w|
How do you determine the Cartesian equation of a plane from the vector equation of a plane?
In the Cartesian equation of a plane, there are no parameters. Hence, to obtain the Cartesian equation of a plane from the vector equation, we eliminate the parameters Combine the vector line (equation for x, y and z) to eliminate the parameters (remember to use all equations)
Is the vector equation of a plane unique?
It is not
Is subtraction of vectors commutative?
Subtraction of vectors is not commutative, i.e. v-w ≠ w-v
A vector can be decomposed into its components in the x and y-direction, and if 3D, z-direction state what is meant by this
That if looking at a vector in a diagram then a vector is composed of the distances in the different axis
How is the general vector equation defined graphically?
The coordinate of the point P(x, y, z) that r points to are the components of r
What is the vector product also called?
The cross product
How can the scalar/dot product be given in the vector's x and y-components?
The dot product can be written ass the sum of the products of the corresponding x and y components - also holds for 3D
What are the properties of the magnitude of a vector?
The last one is only true if the two vectors are parallel
What happens when a vector is multiplied by a scalar?
The scalar multiple changes, the length called the magnitude, of a vector (k≠1), but not its direction except if the scalar is negative (k<0), in which case its direction changes by 180°
What does the scalar/dot product produce?
The scalar/dot product produces a scalar quantity, i.e. a pure number
How can you find the speed of an object?
The speed of an object is defined as the magnitude of its velocity vector
How can the collision of two objects be determined in terms of vectors?
The value for the parameter t, should be the same
Are vector equations of a line unique?
The vector equation of a line is not unique. We could have any other two points on the line so that the actual values of a and b would have been different. However their combination does describe one and only one line.
What can be said about line when they are parallel?
Their directions are the same but they do not share a point
What is the relationship between u×v = -v×u?
They are parallel, but point in opposite directions
What can be said about two lines that intersect?
They have one point in common, the point of intersection, and in 2D non-parallel lines always intersect
What can be said about line when they are skew?
This happens in 3D but not in 2D. The two lines exists in two separate planes. They are not parallel but do not share a single point and so do not intersect, even if their directions are different.
What can be said if two vectors are parallel, in terms of their scalar multiple?
Two vectors are parallel if and only if they are a scalar multiple of each other. For k>0 they point in the same direction and for k<0 they point in opposite directions. If this is true then v×w = |v||w|
Define a vector
Vectors are displacements between two points. They have a size or magnitude, and a direction.
What are unit vectors?
Vectors with a magnitude of 1 |i|=|j|=|k|=1
What are the different solutions to three planes intersecting?
When a≠0 and e≠0
What is the right hand rule to determine the direction of the vector product?
https://www.youtube.com/watch?time_continue=1&v=zGyfiOqiR4s&feature=emb_logo
What is the position vector r, in terms of the position vector of a given point on the plane, a and the normal vector n?
if a is the position vector of a given point on a plane and n is a vector normal to that plane, then the vector r to any point on the plane is given by: r×n=a×n
A vector can be written in i, j, k form/ base vector form, how is this done?
v = xi + yj + xk where x, y and z are the components of the vector and i, j and k are unit vectors.
How is the general vector equation of a line defined?
where b is the direction The coordinates of r is given by: x=a₁+λb₁, y=a₂+λb₂, z=a₃+λb₃