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MH HSC 12 Maths

Vectors

Each concept is a short lesson, then questions until you get three right.

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13 concepts, a short lesson each

  • Vectors0/4
  • Dot product0/3
  • Cross product0/2
  • Triple products0/4

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Vectors in space and their magnitude

a⃗=xi^+yj^+zk^\vec a = x\hat i + y\hat j + z\hat k has ∣a⃗∣=x2+y2+z2|\vec a| = \sqrt{x^2 + y^2 + z^2}; AB→\overrightarrow{AB} = position vector of B minus that of A; a unit vector is a⃗/∣a⃗∣\vec a/|\vec a|

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Vectors0 of 4

  1. Vectors in space and their magnitudea⃗=xi^+yj^+zk^\vec a = x\hat i + y\hat j + z\hat k has ∣a⃗∣=x2+y2+z2|\vec a| = \sqrt{x^2 + y^2 + z^2}; AB→\overrightarrow{AB} = position vector of B minus that of A; a unit vector is a⃗/∣a⃗∣\vec a/|\vec a|Start
  2. Adding vectors and the triangle lawAdd and subtract component by component; around a closed figure the vectors add to 0⃗\vec 0; scalar multiples are parallelStart
  3. Collinear and coplanar vectorsThree points are collinear when AB→=k AC→\overrightarrow{AB} = k\,\overrightarrow{AC}; a vector is a linear combination xa⃗+yb⃗+zc⃗x\vec a + y\vec b + z\vec c of three non-coplanar onesStart
  4. Section formulaInternal division in m:nm : n: mb⃗+na⃗m+n\dfrac{m\vec b + n\vec a}{m + n}; external: mb⃗−na⃗m−n\dfrac{m\vec b - n\vec a}{m - n}; midpoint and centroid as special casesStart

Dot product0 of 3

  1. Dot product and the angle between vectorsa⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ=a1b1+a2b2+a3b3\vec a\cdot\vec b = |\vec a||\vec b|\cos\theta = a_1b_1 + a_2b_2 + a_3b_3; zero means perpendicularStart
  2. Projection of a vectorThe scalar projection of a⃗\vec a on b⃗\vec b is a⃗⋅b⃗∣b⃗∣\dfrac{\vec a\cdot\vec b}{|\vec b|}; the vector projection multiplies it by b^\hat bStart
  3. Direction cosines and direction ratiosl=cos⁡αl = \cos\alpha, m=cos⁡βm = \cos\beta, n=cos⁡γn = \cos\gamma with l2+m2+n2=1l^2 + m^2 + n^2 = 1; ratios a:b:ca : b : c scale to cosines by 1a2+b2+c2\dfrac{1}{\sqrt{a^2 + b^2 + c^2}}Start

Cross product0 of 2

  1. Cross producta⃗×b⃗\vec a\times\vec b is the determinant with rows i^ j^ k^\hat i\ \hat j\ \hat k, a⃗\vec a, b⃗\vec b; it is perpendicular to both, and b⃗×a⃗=−a⃗×b⃗\vec b\times\vec a = -\vec a\times\vec bStart
  2. Areas and unit normals from the cross product∣a⃗×b⃗∣=∣a⃗∣∣b⃗∣sin⁡θ|\vec a\times\vec b| = |\vec a||\vec b|\sin\theta is the parallelogram's area; half of it for a triangle; ±a⃗×b⃗∣a⃗×b⃗∣\pm\dfrac{\vec a\times\vec b}{|\vec a\times\vec b|} are the unit normalsStart

Triple products0 of 4

  1. Scalar triple product[a⃗ b⃗ c⃗]=a⃗⋅(b⃗×c⃗)[\vec a\ \vec b\ \vec c] = \vec a\cdot(\vec b\times\vec c), the 3 by 3 determinant of the components; cyclic swaps keep it, one swap negates itStart
  2. Volume and coplanarity∣[a⃗ b⃗ c⃗]∣|[\vec a\ \vec b\ \vec c]| is the parallelepiped's volume, one sixth of it the tetrahedron's; it is 00 exactly when the three vectors are coplanarStart
  3. Vector triple producta⃗×(b⃗×c⃗)=(a⃗⋅c⃗)b⃗−(a⃗⋅b⃗)c⃗\vec a\times(\vec b\times\vec c) = (\vec a\cdot\vec c)\vec b - (\vec a\cdot\vec b)\vec c, which lies in the plane of b⃗\vec b and c⃗\vec cStart
  4. Identities with triple productsExpand brackets in [ ][\ ], drop terms with a repeated vector: [a⃗+b⃗  b⃗+c⃗  c⃗+a⃗]=2[a⃗ b⃗ c⃗][\vec a + \vec b\ \ \vec b + \vec c\ \ \vec c + \vec a] = 2[\vec a\ \vec b\ \vec c]Start

Practise Vectors

Once the concepts click, try the questions your exam asks.

Textbook exercisesEvery exercise, with model answersPast board questionsAsked from this chapter, with solutions