Unit 1: Vectors and Linear Combinations
Vectors as arrows, the dot product as an angle detector, and span as the whole territory a handful of arrows can reach.
10 topics · 40 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
What a Vector Is
- How it worksHow does listing two numbers in a column turn into an arrow you can actually draw?
- Common mistakeWhy isn't a vector the same thing as a point, even though you draw both as a dot?
- Going furtherWhat changes about a vector's arrow when you slide its tail anywhere else in the plane?
Your turn1 of 2
A vector is drawn as an arrow from the origin to the point (7, 6). If you slide this same arrow so its tail starts at (5, 5) instead, keeping its length and direction unchanged, what is true about the vector it represents?Adding and Scaling Vectors
- How it worksHow does tip-to-tail addition turn two arrows into one combined arrow?
- Worked exampleFind the sum of the vectors (2, 3) and (4, -1).
- Worked exampleFind 3 times the vector (1, -2) and say which direction it points.
- Common mistakeWhy doesn't adding two vectors mean adding their lengths?
Your turn1 of 2
Find the length of the sum of the vectors (3, 2) and (7, -4).Linear Combinations of Vectors
- How it worksHow does picking different weights c and d in cv + dw sweep out every reachable point from two vectors?
- Worked exampleFind the linear combination 2 times (1, 0) plus 3 times (0, 1) and say what point it lands on.
- Worked exampleFind weights c and d so that c times (1, 2) plus d times (3, 1) equals (5, 5).
- Going furtherWhat happens to the reachable points when the two vectors point in exactly the same direction?
Your turn1 of 2
Find the first component of 5 times (2, 5) plus 3 times (-5, -4).The Standard Basis Vectors
- How it worksHow do the standard basis vectors (1, 0) and (0, 1) combine to rebuild any vector in the plane?
- Worked exampleFind how the vector (5, -3) breaks down into the standard basis vectors (1, 0) and (0, 1).
- Going furtherWhat changes about the standard basis vectors when you move from the plane to 3-space?
- Common mistakeWhy isn't (2, 0) and (0, 2) also called a standard basis, even though it spans the same plane?
Your turn1 of 2
The vector (4, 3) breaks down as a times the standard basis vector (1, 0) plus b times (0, 1). Find a - b.The Dot Product and Length
- How it worksHow does multiplying matching entries and adding them up measure how much two vectors point the same way?
- Worked exampleFind the dot product of (1, 2, 3) and (4, -1, 2).
- Worked exampleFind the length of the vector (3, 4).
- Common mistakeWhy isn't the dot product of two vectors itself a vector?
Your turn1 of 2
Find the dot product of (3, 2, 7) and (-4, -7, -5).The Angle Between Two Vectors
- How it worksHow does dividing the dot product by the two lengths turn it into the cosine of the angle between them?
- Worked exampleFind the angle between the vectors (1, 0) and (1, 1).
- Common mistakeWhy doesn't a dot product of zero mean one of the two vectors has to be the zero vector?
- Going furtherWhat happens to the dot product formula the moment two vectors are exactly perpendicular?
Your turn1 of 2
Find the cosine of the angle between the vectors (7, 6) and (8, 3).The Span of a Set of Vectors
- How it worksHow does sweeping every possible combination of two vectors fill in an entire plane?
- Worked exampleFind whether the vector (2, 4) lies in the span of (1, 2).
- Worked exampleShow that the vector (1, 1, 1) can be built by combining (1, 0, 0) and (0, 1, 0).
- Common mistakeWhy doesn't the span of two vectors always turn out to be a full plane?
- Going furtherWhat happens to the span of three vectors in 3-space when one of them is a combination of the other two?
Your turn1 of 2
Two vectors u and v in the plane are given. Under what condition does their span fail to be the entire plane?When Vectors Are Linearly Dependent
- How it worksHow does a vector that already lies in the span of the others get written as a combination of them?
- Worked exampleShow that the vectors (1, 2), (2, 4), and (0, 1) are linearly dependent.
- Common mistakeWhy doesn't a set of dependent vectors have to include the zero vector?
- Going furtherWhat happens to the span of a set of vectors once one of them turns out to be a combination of the rest?
Your turn1 of 2
Find ad - bc for the vectors (4, 3) and (8, -4). This value is zero exactly when the two vectors are dependent (parallel).Vectors in n Dimensions
- How it worksHow do length, addition, and the dot product all still make sense once a vector has far more than 3 entries?
- Worked exampleFind the length of the vector (1, 2, 2, 4) in 4-dimensional space.
- Common mistakeWhy doesn't a vector need to live in a space you can physically picture to count as a fully valid vector?
- Going furtherWhat happens to the dot product formula when two vectors live in 100-dimensional space instead of 3?
Your turn1 of 2
Find the length of the vector (4, 3, 8, -4) in 4-dimensional space.Lines and Planes in Space
- How it worksHow does a point plus a direction vector times a parameter trace out every point on a line?
- Worked exampleFind the parametric equation of the line through the point (1, 0, 2) in the direction (2, 1, -1).
- Going furtherWhat changes about a plane's equation when its normal vector is doubled in length?
- Ties togetherHow does a plane through the origin turn out to be exactly the span of two vectors?
Your turn1 of 2
A line passes through the point (4, 3, 8) in the direction (2, -5, -4). Find the x-coordinate of the point on this line at parameter t = 4.