Porisma

Vectors: magnitude, direction and products

Learn one idea at a time

Choose a unit. Each has worked examples and practice.

  1. Read vector types and notation

    Distinguish length, direction, position and displacement.

  2. Add, subtract and scale vectors

    Use path addition, reverse a vector and scale a unit direction.

  3. Work with Cartesian vectors

    Connect coordinates, basis vectors, translation and length in two and three dimensions.

  4. Prove collinearity with a common point

    Test a scalar relation and identify the common endpoint.

  5. Divide a segment in a stated ratio

    Choose the point order before applying internal weights.

  6. Compare coefficients in two independent directions

    Describe one intersection using two lines and compare independent coefficients.

  7. Prove the centroid lies on all three medians

    Use two median equations to prove the common point lies on the third.

  8. Read the scalar product and its laws

    Keep lengths, signs and arrow conventions explicit.

  9. Calculate scalar products from Cartesian components

    Expand using an orthonormal basis and use the result in equations.

  10. Find the requested angle without reversing one vector

    Choose the common vertex and round the signed inverse cosine correctly.

  11. Relate a direction to the positive coordinate axes

    Use unit-axis vectors and preserve component signs.

  12. Resolve parallel and perpendicular components

    Separate nonnegative length from signed vector components.

  13. Use the cross-product direction and laws

    Track the right-hand normal, signs and nonzero conditions.

  14. Calculate and check a Cartesian cross product

    Expand with cyclic basis rules and verify both dot products.

  15. Use a cross-product magnitude for triangle area

    Use common-vertex side vectors for triangle and parallelogram areas, then derive the position-vector formula.

  16. Compare parallel and perpendicular lengths

    Choose a scalar length formula and compare the six product properties.

  17. Connect vectors with motion and work

    Explain three physical settings and the force component that contributes to work.

  18. Review mastery and choose the next practice

    Check all seventeen source objectives and choose practice by method.

Read or print the complete chapter

By the end, you can

  • Use displacement, magnitude and unit vectors
  • Interpret dot and cross products geometrically
  • Apply vectors to collinearity and projection

18 focused units · an hour and a half for the full chapter. Allow extra time for practice.

The central idea

A vector combines magnitude and direction. The displacement from A to B is b−a\mathbf b-\mathbf a. The dot product a⋅b=∣a∣∣b∣cos⁡θ\mathbf a\cdot\mathbf b=|\mathbf a||\mathbf b|\cos\theta measures alignment; a zero dot product means perpendicular nonzero vectors. The cross product is perpendicular to both vectors and its magnitude gives parallelogram area.

Vector vocabulary and normalisation

A directed segment represents a vector: its length is the modulus ∣a∣|\mathbf a| and its arrow gives direction. The zero vector has zero magnitude. Two nonzero vectors are parallel precisely when a=kb\mathbf a=k\mathbf b for a nonzero real k. The vector −a-\mathbf a has the same magnitude and opposite direction.

For a≠0\mathbf a\ne\mathbf0, a^=a/∣a∣\widehat{\mathbf a}=\mathbf a/|\mathbf a| has unit magnitude and a=∣a∣a^\mathbf a=|\mathbf a|\widehat{\mathbf a}. A vector of magnitude 6 in that same direction is 6a^6\widehat{\mathbf a}. The zero vector cannot be normalised. These definitions extend the magnitude/direction view to points, lines and planes.

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Worked example

For a=(3,4,0)\mathbf a=(3,4,0), ∣a∣=5|\mathbf a|=5 and its unit vector is (3/5,4/5,0)(3/5,4/5,0). With b=(4,−3,0)\mathbf b=(4,-3,0), the dot product is zero, so the vectors are perpendicular.

A unit vector has magnitude 1, but its components need not each be 1.

Check your understanding

Let p=(2,−1,2)\mathbf p=(2,-1,2) and b=(0,3,0)\mathbf b=(0,3,0). Resolve p\mathbf p into components parallel and perpendicular to b\mathbf b. Give the signed scalar component along b\mathbf b, the projection length, and the area of the triangle whose two sides from one vertex are p\mathbf p and b\mathbf b. Explain why these quantities cannot all be read directly from p⋅b\mathbf p\cdot\mathbf b.

Practice questions

52 examples and practice questions, in 9 sections.

More practice

Identify whether the requested result is a scalar, vector, length or area. Predict its sign and geometric meaning before calculating, then check it independently.

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