The scalar product and its uses in space · Grade 12 · 5 units
In The scalar product and its uses in space (Grade 12 · 5 units) you practise 3 skills: The laws of the scalar product and the parallelogram law · Perpendicular lines in the cube · The three perpendiculars theorem. The scalar product of two vectors is a number: their magnitudes times the cosine of the angle between them. It keeps the commutative and the distributive laws, so brackets are expanded just as they are for numbers. A vector squared equals its magnitude squared, and that gives the length of a sum or a difference: expand the square, then put in the magnitudes and the product. Add the square of the sum to the square of the difference and the products cancel. That is the parallelogram law: the squares of the diagonals add up to the squares of the four sides.
Two vectors other than the zero vector are perpendicular exactly when their scalar product is zero. This also tests skew lines, which never meet: the angle between two lines is the angle between their vectors. In a cube, for example, a diagonal of the base is perpendicular to the other diagonal of the top face, although they never meet. A line is perpendicular to a plane when it is perpendicular to two lines of the plane that are not parallel. One line is not enough, and neither are two parallel lines. The three perpendiculars theorem links a slant segment to its projection on a plane. If a segment is perpendicular to a plane, and the projection of a slant segment is perpendicular to a line of the plane, then the slant segment is perpendicular to that line too. The converse holds as well: if the slant segment is perpendicular to a line of the plane, so is its projection. The theorem finds lengths along a chain of right triangles, with Pythagoras’ theorem in each. A model proof of the three perpendiculars theorem: the segment SO is perpendicular to a plane at O, the points A and B lie in the plane, and OA is perpendicular to AB. The vector SA is the sum of the vectors SO and OA. So the scalar product of SA with AB is the sum of two products. The product of SO with AB is zero, since SO is perpendicular to every line of the plane. The product of OA with AB is zero, since OA is perpendicular to AB. So the whole product is zero, and SA is perpendicular to AB.
Tip: Before you conclude that a line is perpendicular to a plane, check that the two lines of the plane are not parallel.
What you learn
The laws of the scalar product and the parallelogram law
For example: The magnitudes |u⃗| = 3 and |v⃗| = 9 are given, and the angle between the vectors is 60° . What is the magnitude |u⃗ + v⃗| , in its simplest form?
Answer and solution
The answer: 3√13
Twice the scalar product is 2 × 3 × 9 × ½ , which is 27 . So |u⃗ + v⃗|² = 9 + 81 + 27 , that is, |u⃗ + v⃗|² = 117 , so |u⃗ + v⃗| = 3√13 .
Perpendicular lines in the cube
For example: In the cube ABCDA′B′C′D′ , which face diagonal is perpendicular to the diagonal BD ?
- AD′
- A′B
- B′C
- A′C′
Answer and solution
The answer: A′C′
Place the cube on axes with edges of length 1 , the corner A at the origin and the edges AB , AD and AA′ along the axes. Then BD⃗ = (−1, 1, 0) and A′C′⃗ = (1, 1, 0) , and their scalar product is zero. So the perpendicular diagonal is A′C′ . Each of the other diagonals has a product with BD⃗ that is not zero.
The three perpendiculars theorem
For example: In the box ABCDA′B′C′D′ , which segment is perpendicular to the edge DC ?
- A′C
- A′B
- AC
- A′D
Answer and solution
The answer: A′D
The edge A′A is perpendicular to the base, and the projection of A′D on the base is AD , which is perpendicular to DC because the base is a rectangle. By the three perpendiculars theorem, the segment A′D is perpendicular to DC . The projections of the other segments are not perpendicular to the edge.
Not on the site yet
Parts of this topic that are not built yet: Comparing the vector tool with trigonometry.
The exercises are made by rules and the answers are worked out in code, so there are new exercises every time. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 5 units (grades 10 to 12).