Vectors
IMPORTANT You will need to have a WolframAlpha Pro account to use Wolfram effectively.
Wolfram has it’s own documentation.
Displaying
Typing vector into the search bar followed by your vector in Xi+Yj notation will have it displayed. They are displayed in coordinate geometry as linear combinations of unit vectors. It also suggests more information like their lengths and normalized vectors.
You can use the same and notation as with functions to display more than one vector, as in the above link.
Vector addition is as simple as putting a + between two vectors: vector (2i - 4j + 2k) + (3i +3j) . You can omit or include the coordinate notation.
Products
Dot and cross products are as simple as writing (your-vector-1).(your-vector-2) or (your-vector-1) dot (your-vector-2), and (your-vector-1)x(your-vector-2) or (your-vector-1) cross (your-vector-2). For the cross product you can visualize the resulting vector, which is normal to the plane describe for the vectors that are multiplied.
For the triple scalar product (e.g. to find the volume of a parallelpiped), you have to do the operations seperately - but they can be in the same query.
Describe a line
Select a initial point, the line direction, and a scalar parameter. The resulting vector describes a point belonging to the line. Here there is a link to a Wolfram demonstration that may help you to visualize the concept of a line described by vectors.
Describe a plane
Using vectors to describe a plane - select a initial point, two line directions, and two scalar parameters. The resulting vector describes a point belonging to the plane. Here there is a link to a Wolfram demonstration that may help you to visualize the concept of a plane described by vectors.
Closest Approach
While you can’t calculate the closest approach of two vector lines directly in WolframAlpha, there are some great resources for this out there. For example, the Keisan calculator will tell you the closest distance between two vector lines you input.