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Vectors
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A Geometry & Algebra set, untimed
Named traps
8 ways this goes wrong.
Each is a named misconception the questions are built to catch. Open one for the move that tempts, and the fix.
- Cross term dropped in |a+b|Forgets the 2(a.b) term when finding the magnitude of a vector sum.→
- Parallel set components equal not proportionalSets components equal instead of proportional for a parallel-vector condition.→
- Parallel vs perpendicular conditionUses the dot-product (perpendicular) condition when parallel/collinear is required, or vice versa.→
- Polygon vector relation misusedMisuses a hexagon's diagonal/side relations or picks the wrong base-vector combination.→
- Scalar vs projection vectorReports the scalar projection when the projection vector is required (or drops the orthogonal cross term).→
- Vector subtraction order reversedComputes A-B instead of B-A (or adds instead of subtracts) for vector AB.→
- Wrong dot product set to zeroSets the wrong dot product to zero when applying a perpendicularity condition with |a|=k|b|.→
- Wrong included angle usedUses cos of the wrong angle (e.g. 60 vs 120) in the law-of-cosines magnitude.→
Key formulas
6- Coordinates of ABB − A (component-wise)magnitude |a| = √ax2 + ay2 (+ az2)
- Dot producta·b = |a||b| cos θ = ax bx + ay by (+ az bz)
- Angle between vectorscos θ = (a·b)/(|a||b|)
- Perpendicular: a·b = 0Parallel/collinear (2D): ax by − ay bx = 0 (components proportional)
- Magnitude of a sum|a+b|2 = |a|2 + 2(a·b) + |b|2
- Projection vector of a onto b((a·b)/|b|2) bscalar projection (a·b)/|b|
Shaky underneath?
If the concept underneath still feels loose, patch it here before coming back to CSCA questions.
FoundationsPrecalculus bridgeAlgebra 2 foundations