Pure Mathematics · 3rd secondary · First Term

Algebra - Geometry and measurement in two and three dimensions — Vector multiplication

About this lesson

This unit covers the multiplication of vectors in two and three dimensions, focusing on scalar (dot) and vector (cross) products. It begins by defining the scalar product of two vectors using the angle between them, then explores its algebraic and geometric properties, including perpendicularity conditions and the component (projection) of a vector in the direction of another. It then introduces the vector product, its properties, the right-hand rule for direction, and its geometric meaning as the area of a parallelogram or triangle. The unit also covers the scalar triple product and its geometric interpretation as the volume of a parallelepiped. Later sections shift to straight lines and planes in space, covering equations of lines (vector, parametric, Cartesian) and planes (general, standard, intercepts), angles between lines and planes, parallelism and perpendicularity conditions, and distances between points, lines, and planes. Throughout, examples and exercises use coordinate systems and unit vectors (i, j, k) to illustrate concepts.

Main topics in this lesson

  • Scalar product of two vectors
  • Vector product of two vectors
  • Component and projection of a vector
  • Scalar triple product
  • Equations of straight lines in space
  • Equations of planes in space
  • Angles between lines and planes
  • Distances in space

Key terms and vocabulary

  • Scalar product
  • Dot product
  • Vector product
  • Cross product
  • Component
  • Projection
  • Unit vector
  • Right hand rule
  • Scalar triple product
  • Direction vector
  • Direction angles
  • Direction cosines
  • Direction ratios
  • Vector equation
  • Parametric equations
  • Cartesian equation
  • General equation
  • Standard form
  • Parallel straight lines
  • Perpendicular straight lines
  • Intersecting straight lines
  • Skew straight lines
  • Parallel planes
  • Perpendicular planes
  • Intersecting planes
  • Angle
  • Perpendicular distance

Questions and answers on Algebra - Geometry and measurement in two and three dimensions — Vector multiplication - practice and revision

Multiple choice questions on this lesson in Pure Mathematics for 3rd secondary, First Term, with practice exercises for revision and exam preparation.

1

إذا كان أ = 4س + 2ص، ب = س + 3ص، فإن مساحة المثلث الذي ضلعاه المتجهان أ و ب تساوي:

  • 1 5
  • 2 10
  • 3 20
  • 4 2.5
2

المتجهان أ = 3س − 2ص + ع، ب = −6س + 4ص − 2ع يكونان:

  • 1 متعامدين
  • 2 متوازيين
  • 3 متساويين
  • 4 متباعدين
3

المستقيم في الفضاء المار بالنقطة (1، 2، 3) ومتجه اتجاهه (2، 1، −1) تكون معادلاته البارامترية:

  • 1 س = 1 + 2ن، ص = 2 + ن، ع = 3 − ن
  • 2 س = 2 + ن، ص = 1 + 2ن، ع = −1 + 3ن
  • 3 س = 1 − 2ن، ص = 2 + ن، ع = 3 + ن
  • 4 س = 1 + ن، ص = 2 + ن، ع = 3 + ن
4

المتجه أ = 2س + 3ص + 6ع، فإن جيوب الاتجاه للمتجه أ هي:

  • 1 2/7، 3/7، 6/7
  • 2 2/11، 3/11، 6/11
  • 3 2، 3، 6
  • 4 2/5، 3/5، 6/5
5

إذا كان أ = س + ص، ب = ص + ع، جـ = ع + س، فإن الضرب الثلاثي القياسي أ · (ب × جـ) يساوي:

  • 1 2
  • 2 1
  • 3 −2
  • 4 0
6

إذا كان أ = 2س + 3ص + ع، ب = س − ص + 2ع، فإن حاصل الضرب المتجهي أ × ب يساوي:

  • 1 7س − 3ص − 5ع
  • 2 7س + 3ص − 5ع
  • 3 −7س + 3ص + 5ع
  • 4 7س − 3ص + 5ع

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