Practice: Stepping up to 3D
Self-check
Section titled “Self-check”Six short questions. Answer each one in your head (or on paper) before opening the collapsible. Trying to retrieve the answer is where the learning sticks; rereading feels productive but does much less.
1. Going from 2D to 3D, what actually changes about the machinery?
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Almost nothing. You add a third basis vector (k-hat), the matrix gains a third column, each vector and column gains a third entry, and the matrix-vector product gains a third term. The definition of linear, the basis-vector trick, and composition all carry over unchanged.
2. What are the three standard basis vectors in 3D?
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i-hat = [1, 0, 0] (along x), j-hat = [0, 1, 0] (along y), and k-hat = [0, 0, 1] (along z). Every 3D vector is a combination of these three, for example [3, 4, 5] = 3 · i-hat + 4 · j-hat + 5 · k-hat.
3. How many landing spots determine a 3D linear transformation, and how are they stored?
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Three: where i-hat, j-hat, and k-hat land. They are stored as the three columns of a 3x3 matrix, each column a 3D vector. Knowing those three reconstructs where every vector goes.
4. What is the matrix-vector product M · v for a 3x3 matrix and v = [x, y, z]?
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M · v = x · (first column) + y · (second column) + z · (third column). It is the same scale-and-add as 2D, now with three terms instead of two.
5. Why must you name an axis when you talk about rotation in 3D?
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Because “rotation” is no longer a single transformation in 3D. Rotating about the z-axis, the y-axis, and the x-axis are three different transformations with three different matrices, each leaving a different direction fixed. In 2D there was only one way to rotate, so no axis needed naming.
6. What does the unit cube become under a 3x3 matrix, and how do you read it?
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It becomes a parallelepiped (a slanted box). Plot the three columns as arrows from the origin; those are the three edges of the transformed box. The shape tells you the whole transformation: a box (stretch), a spun cube (rotation), a slanted box (shear), or a nearly flat shape (collapse toward a plane).
Try it yourself, part 1: apply a 3x3 matrix
Section titled “Try it yourself, part 1: apply a 3x3 matrix”Apply each matrix to v = [2, 1, 3] using x · col1 + y · col2 + z · col3. About 8 minutes, pen and paper.
- a) Uniform scale by 2:
[[2, 0, 0], [0, 2, 0], [0, 0, 2]] - b) Rotate 90 degrees about the z-axis:
[[0, -1, 0], [1, 0, 0], [0, 0, 1]] - c) Rotate 90 degrees about the x-axis:
[[1, 0, 0], [0, 0, -1], [0, 1, 0]]
Check your work
- a) Columns
[2,0,0],[0,2,0],[0,0,2].2 · [2,0,0] + 1 · [0,2,0] + 3 · [0,0,2] = [4, 2, 6]. Every coordinate doubled. - b) Columns
[0,1,0],[-1,0,0],[0,0,1].2 · [0,1,0] + 1 · [-1,0,0] + 3 · [0,0,1] = [0,2,0] + [-1,0,0] + [0,0,3] = [-1, 2, 3]. The x and y parts rotate (like the 2D quarter turn); the height 3 rides along untouched. - c) Columns
[1,0,0],[0,0,1],[0,-1,0].2 · [1,0,0] + 1 · [0,0,1] + 3 · [0,-1,0] = [2,0,0] + [0,0,1] + [0,-3,0] = [2, -3, 1]. This time x is the fixed axis; y and z swap with a sign flip.
Try it yourself, part 2: read the unit cube’s image
Section titled “Try it yourself, part 2: read the unit cube’s image”For each matrix, read the three columns (where i-hat, j-hat, k-hat land) and name what the unit cube becomes. About 6 minutes.
- a)
[[2, 0, 0], [0, 1, 0], [0, 0, 1]] - b)
[[0, -1, 0], [1, 0, 0], [0, 0, 1]] - c)
[[1, 0, 1], [0, 1, 0], [0, 0, 1]] - d)
[[1, 0, 0], [0, 1, 0], [0, 0, 0]]
Show answer
- a) Columns
[2,0,0],[0,1,0],[0,0,1]: a box stretched to 2 units wide along x, still 1 tall and 1 deep. No slant. - b) Columns
[0,1,0],[-1,0,0],[0,0,1]: the cube stays a cube, spun a quarter turn about the z-axis (all columns still unit length and perpendicular). - c) Columns
[1,0,0],[0,1,0],[1,0,1]:k-hattips toward+x, so the cube slants into a parallelepiped (a shear). - d) Columns
[1,0,0],[0,1,0],[0,0,0]: the third column is the zero vector, so depth collapses. The cube flattens onto the x-y plane (a unit square). The three columns are coplanar, the warning sign that the transformation is crushing 3D space down to a plane.
Flashcards
Section titled “Flashcards”Ten cards. Click any card to reveal the answer. Use the Print flashcards button to lay out the full set as one card per page, ready to print or save as a PDF for offline review.
Q. Going from 2D to 3D, what changes about the machinery?
Almost nothing: one more basis vector (k-hat), one more column, one more entry per vector, one more term in M·v. The definition of linear, the basis-vector trick, and composition all carry over unchanged.
Q. What are the three standard basis vectors in 3D?
i-hat = [1, 0, 0], j-hat = [0, 1, 0], k-hat = [0, 0, 1], pointing along x, y, and z. Every 3D vector is a combination of these three.
Q. How many landing spots determine a 3D transformation, and how are they stored?
Three: where i-hat, j-hat, and k-hat land, stored as the three columns of a 3x3 matrix. Those three reconstruct where every vector goes.
Q. What is M·v for a 3x3 matrix and v = [x, y, z]?
M · v = x · col1 + y · col2 + z · col3: the same scale-and-add as 2D, with three terms instead of two. The columns are the transformed basis vectors.
Q. Why must you name the axis when rotating in 3D?
Because rotation is not a single transformation in 3D. Rotating about the z, y, and x axes are three different transformations with three different matrices, each leaving a different direction fixed.
Q. What does the unit cube become under a 3x3 matrix?
A parallelepiped (a slanted box) whose three edges from the origin are the three columns of the matrix. Plot the columns and the box shows the whole transformation.
Q. What is a parallelepiped?
The 3D analogue of a parallelogram: the slanted box the unit cube becomes under a linear transformation. Its three edges from the origin are the matrix’s three columns.
Q. What does a diagonal 3x3 matrix do?
It scales each axis independently by the diagonal entries. For example [[2,0,0],[0,3,0],[0,0,0.5]] doubles x, triples y, and halves z, turning the cube into a rectangular box with no slant.
Q. What does it mean if the three columns of a 3x3 matrix are nearly coplanar?
The parallelepiped squashes nearly flat: the transformation is collapsing 3D space toward a plane or a line. If they are exactly coplanar, the cube flattens completely (a sign you will meet again as a zero determinant).
Q. How does the 2D-to-3D leap relate to high-dimensional spaces like a 768-d embedding?
It is the same leap. A 768-dimensional space just has 768 basis directions, vectors with 768 entries, and matrices with 768 columns, following the exact rules used on [3, 4, 5]. High-dimensional means long list, not magic; 3D is the last one you can draw.