A vector is represented geometrically as an arrow.
In column vector notation, a vector AB from A(x1,y1) to B(x2,y2) is expressed as AB=(x2−x1y2−y1), where x2−x1 and y2−y1 are its components
What do the components of a vector represent in column vector notation?
Movement along axes
A vector is a quantity with both magnitude and direction
Vectors build upon coordinate geometry by giving direction and magnitude to coordinate differences.
What is a position vector?
Origin to a point
A direction vector indicates the direction of a vector without specifying its location
Position and direction vectors are essential for describing points and directions in vector geometry.
What is the vector equation of a line?
r=a+td
Match the terms in the vector equation of a line with their meanings:
r ↔️ Position vector of any point on the line
a ↔️ Position vector of a specific point on the line
d ↔️ Direction vector of the line
t ↔️ Scalar parameter
In the vector equation of a line, the direction vector indicates the direction
Vectors extend coordinate geometry by adding direction and magnitude to coordinate differences.
What is the position vector of a point P(x,y,z)?
OP=xyz
What do the components of the vector (34) indicate in terms of movement?
3 units along the x-axis and 4 units along the y-axis
Vectors build upon coordinate geometry by giving direction and magnitude
A position vector points from the origin to a specific point in space.
What does a direction vector indicate about a vector?
Direction
Understanding direction vectors helps describe the orientation of lines and planes.
What is the vector equation of a line?
r=a+td
In the vector equation of a line, r represents the position vector of any point on the line
What does the direction vector d indicate in the vector equation of a line?
The direction of the line
Match the component of the vector equation of a line with its description:
r ↔️ Position vector of any point on the line
a ↔️ Position vector of a specific point on the line
d ↔️ Direction vector of the line
t ↔️ Scalar parameter
What is the vector equation of a line passing through (1,2) with direction (3−1)?
r=(12)+t(3−1)
The vector equation of a plane uses a normal vector and a position vector.
What is the vector equation of a plane?
n⋅(r−a)=0
In the vector equation of a plane, n represents the normal vector to the plane
The normal vector to a plane is perpendicular to the plane.
What is the vector equation of a plane with normal vector 21−1 and a point A(1,2,3)?
21−1⋅(xyz−123)=0
The simplified vector equation of a plane in the example is 2x + y - z = 1</latex>.
How do you determine if a point lies on a line using its vector equation?
Solve for t and check consistency
A point lies on a plane if substituting its coordinates into the plane's vector equation satisfies the equation.
What two properties define a vector?
Magnitude and direction
Match the term with its description:
Magnitude ↔️ Length of the vector
Direction ↔️ Orientation of the vector
Column vector notation ↔️ Representation of a vector using components
What is the vector AB for points A(1,2) and B(4,6)?
AB=(34)
What does a position vector define about a point in space?
Location
The vector equation of a line is r=a+td.
What does a represent in the vector equation of a line?
Position vector of a specific point
The vector equation of a plane is n⋅(r−a)=0.
What does n represent in the vector equation of a plane?