Common Core · High school
HSN-VM.B.4
High school — Add and subtract vectors.
A note on this mapping: HSN-VM.B.4 is a (+) standard — beyond the college- and career-ready threshold, i.e. precalculus rather than Algebra II.
The 5 skills that cover it
- VectorsA quantity with a size and a direction: write a vector in component form, find its magnitude and direction angle, add, subtract and scale vectors, build one from a magnitude and an angle, and find the unit vector that points the same way.Read the lesson →
- Vectors in n DimensionsA vector stops being an arrow on a page and becomes a list of numbers of any length: add and scale vectors in three, four or n dimensions, measure their length, find the unit vector along one, take a dot product to test whether two are perpendicular, and write a line in space as a point plus a direction.Read the lesson →
- Linear CombinationsBuild one vector out of others: evaluate a combination such as 3u - 2v, work backwards to find the weights that reach a target, decide whether a target can be reached at all, and see that a matrix times a vector is nothing more than a combination of the columns.Read the lesson →
- SpanEverywhere a set of vectors can reach: decide whether a vector lies in a span, recognise a span as the origin, a line, a plane or the whole space, find the value that puts a vector inside one, and say how many vectors it takes to span n dimensions.Read the lesson →
- Vector Spaces & SubspacesWhich sets of vectors are worlds in their own right: check that a set contains the zero vector and is closed under addition and scaling, tell a plane through the origin from one that misses it, name the rule a set breaks, and give the dimension of the subspace that survives.Read the lesson →
What a student should be able to do
- Write the component form of a vector between two points, and find its magnitude.
- Find the direction angle of a vector, and the components of a vector from its magnitude and direction.
- Add and subtract vectors, multiply by a scalar, and simplify a linear combination such as 3u - 2v.
- Find a unit vector in the direction of a given vector, and convert between component and i, j notation.
- Add, subtract and scale vectors in three or more dimensions, component by component.
- Find the magnitude of a vector in n dimensions, the distance between two points, and the unit vector in a given direction.
- Compute the dot product in n dimensions, decide whether two vectors are orthogonal, and find the value that makes them so.
- Write the vector equation of a line in space and find the point at a given parameter.
- Evaluate a linear combination of two or three vectors.
- Find the weights that write one vector as a combination of others, by solving the system they give.
- Decide whether a vector can be written as a combination of a given set, and say why not when it cannot.
- Read a matrix-vector product as a linear combination of the columns of the matrix.
- Decide whether a vector lies in the span of a given set, and find the value that puts it there.
- Describe the span of one, two or three vectors as a point, a line, a plane or the whole space.
- Decide whether the columns of a matrix span the whole space, and how many vectors that needs.
- Give the equation of the plane a span fills.
- Decide whether a set of vectors is a subspace, and name the rule it breaks when it is not.
- Tell a line or plane through the origin from one that does not pass through it.
- Find the value that puts a vector inside a given subspace.
- Give the dimension of a stated subspace, and say what the intersection of two subspaces is.