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Lesson notes Vectors and notation A vector has both magnitude (size) and direction . Written as a column \begin{pmatrix} a \\ b \end{pmatrix}, or \vec{AB}, or in bold \mathbf{a}. \vec{AB} goes from A to B. Its negative reverses direction: \vec{BA} = -\vec{AB}. The zero vector has magnitude 0 and no defined direction. Statements about parallel directions here assume nonzero vectors. Vector magnitude from components Vector magnitude from components a = (3, 0) b = (0, 4) a + b = (3, 4) A B C Operations with vectors Add vectors by adding components: \begin{pmatrix} 2 \\ 3 \end{pmatrix} + \begin{pmatrix} 1 \\ -4 \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}.Multiply by a scalar to scale: 3\begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}.The magnitude of \begin{pmatrix} a \\ b \end{pmatrix} is \√{a2 + b2 }. For \begin{pmatrix} 3 \\ 4 \end{pmatrix} it is 5. Parallel vectors Two nonzero vectors are parallel if one is a nonzero scalar multiple of the other. Positive multiples point the same way; negative multiples point in opposite directions. Geometry with vectors Use the triangle rule: \vec{AC} = \vec{AB} + \vec{BC}. Route to a point by any path along known vectors. The midpoint M of AB satisfies \vec{AM} = \frac{1}{2}\vec{AB}. Vectors prove that points are collinear (on a straight line): if \vec{AB} is a multiple of \vec{AC} then A, B and C are collinear. Adding vectors head to tail Adding vectors head to tail a = (3, 1) b = (1, 3) a + b = (4, 4) A B C Slides Sign up free to view the lesson slides
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Practice questions Free preview — 8 of 32 questions. Sign up to see them all. 1. What two properties does a vector have?
Easy A size onlyB magnitude and directionC direction onlyD speed and time2. What is 2\begin{pmatrix} 3 \\ -1 \end{pmatrix}?
Easy A \begin{pmatrix} 6 \\ -2 \end{pmatrix}B \begin{pmatrix} 5 \\ 1 \end{pmatrix}C \begin{pmatrix} 6 \\ -1 \end{pmatrix}D \begin{pmatrix} 3 \\ -2 \end{pmatrix}3. What is \begin{pmatrix} 4 \\ 1 \end{pmatrix} + \begin{pmatrix} 2 \\ 5 \end{pmatrix}?
Easy A \begin{pmatrix} 6 \\ 6 \end{pmatrix}B \begin{pmatrix} 8 \\ 5 \end{pmatrix}C \begin{pmatrix} 2 \\ -4 \end{pmatrix}D \begin{pmatrix} 6 \\ 4 \end{pmatrix}4. What is \begin{pmatrix} 5 \\ 2 \end{pmatrix} - \begin{pmatrix} 1 \\ 4 \end{pmatrix}?
Easy A \begin{pmatrix} 4 \\ -2 \end{pmatrix}B \begin{pmatrix} 6 \\ 6 \end{pmatrix}C \begin{pmatrix} 4 \\ 2 \end{pmatrix}D \begin{pmatrix} -4 \\ 2 \end{pmatrix}5. What is -\begin{pmatrix} 2 \\ -3 \end{pmatrix}?
Easy A \begin{pmatrix} -2 \\ 3 \end{pmatrix}B \begin{pmatrix} 2 \\ 3 \end{pmatrix}C \begin{pmatrix} -2 \\ -3 \end{pmatrix}D \begin{pmatrix} 3 \\ -2 \end{pmatrix}6. If \vec{AB} = \mathbf{a}, what is \vec{BA}?
Easy A \mathbf{a}B -\mathbf{a}C 2\mathbf{a}D 07. A has coordinates (1, 2) and B (4, 6). What is \vec{AB}?
Easy A \begin{pmatrix} 3 \\ 4 \end{pmatrix}B \begin{pmatrix} 5 \\ 8 \end{pmatrix}C \begin{pmatrix} -3 \\ -4 \end{pmatrix}D \begin{pmatrix} 4 \\ 3 \end{pmatrix}8. What is the magnitude of \begin{pmatrix} 6 \\ 8 \end{pmatrix}?
Easy More questions 9. Which vector is parallel to \begin{pmatrix} 2 \\ 3 \end{pmatrix}?
Medium A \begin{pmatrix} 4 \\ 6 \end{pmatrix}B \begin{pmatrix} 3 \\ 2 \end{pmatrix}C \begin{pmatrix} 2 \\ 4 \end{pmatrix}D \begin{pmatrix} 6 \\ 8 \end{pmatrix}10. If \mathbf{a} = \begin{pmatrix} 1 \\ 2 \end{pmatrix} and \mathbf{b} = \begin{pmatrix} 3 \\ -1 \end{pmatrix}, what is 2\mathbf{a} + \mathbf{b}?
Medium A \begin{pmatrix} 5 \\ 3 \end{pmatrix}B \begin{pmatrix} 4 \\ 1 \end{pmatrix}C \begin{pmatrix} 5 \\ 1 \end{pmatrix}D \begin{pmatrix} 7 \\ 3 \end{pmatrix}11. What is the magnitude of \begin{pmatrix} 5 \\ 12 \end{pmatrix}?
Medium 12. In triangle ABC, \vec{AB} = \mathbf{a} and \vec{BC} = \mathbf{b}. What is \vec{AC}?
Medium A \mathbf{a} + \mathbf{b}B \mathbf{a} - \mathbf{b}C \mathbf{b} - \mathbf{a}D \mathbf{a}\mathbf{b}13. M is the midpoint of AB and \vec{AB} = \mathbf{a}. What is \vec{AM}?
Medium A \frac{1}{2}\mathbf{a}B 2\mathbf{a}C -\mathbf{a}D \mathbf{a}14. What is \vec{AC} if \vec{AB} = \mathbf{a} and \vec{CB} = \mathbf{b}?
Medium A \mathbf{a} - \mathbf{b}B \mathbf{a} + \mathbf{b}C \mathbf{b} - \mathbf{a}D -\mathbf{a} - \mathbf{b}15. \vec{OA} = \mathbf{a}, \vec{OB} = \mathbf{b}. What is \vec{AB}?
Medium A \mathbf{b} - \mathbf{a}B \mathbf{a} - \mathbf{b}C \mathbf{a} + \mathbf{b}D -\mathbf{a} - \mathbf{b}16. \vec{OA} = \mathbf{a}, \vec{OB} = \mathbf{b}, and M is the midpoint of AB. What is \vec{OM}?
Medium A \frac{1}{2}(\mathbf{a} + \mathbf{b})B \frac{1}{2}(\mathbf{b} - \mathbf{a})C \mathbf{a} + \mathbf{b}D \frac{1}{2}\mathbf{a}17. \vec{AB} = \begin{pmatrix} 2 \\ 4 \end{pmatrix} and \vec{AC} = \begin{pmatrix} 6 \\ 12 \end{pmatrix}. What can you say about A, B and C?
Medium A collinearB they form a right angleC AB is longerD no relation18. For which k are \begin{pmatrix} k \\ 6 \end{pmatrix} and \begin{pmatrix} 2 \\ 3 \end{pmatrix} parallel?
Medium 19. What does \begin{pmatrix} 3 \\ -2 \end{pmatrix} represent as a movement?
Easy A 3 right, 2 downB 3 left, 2 upC 3 up, 2 rightD 2 left, 3 down20. If \mathbf{a} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}, what is 3\mathbf{a}?
Easy A \begin{pmatrix} 6 \\ 3 \end{pmatrix}B \begin{pmatrix} 5 \\ 4 \end{pmatrix}C \begin{pmatrix} 6 \\ 1 \end{pmatrix}D \begin{pmatrix} 2 \\ 3 \end{pmatrix}21. Two nonzero vectors are parallel when:
Medium A one is a scalar multiple of the otherB they have equal sumsC they are equal in magnitudeD they are at right angles22. The route from A to C via B shows:
Medium A \vec{AC} = \vec{AB} + \vec{BC}B \vec{AC} = \vec{AB} - \vec{BC}C \vec{AC} = \vec{BC}D \vec{AC} = 023. A vector has both magnitude and direction.
Easy True or false?
24. \vec{BA} = -\vec{AB}.
Easy True or false?
25. The magnitude of \begin{pmatrix} 3 \\ 4 \end{pmatrix} is 7.
Easy True or false?
26. Nonzero parallel vectors are scalar multiples of each other.
Easy True or false?
27. For distinct A, B and C, if AB and AC are parallel vectors, then A, B and C are collinear.
Medium True or false?
28. Which are parallel to \begin{pmatrix} 1 \\ 2 \end{pmatrix}?
Medium A \begin{pmatrix} 2 \\ 4 \end{pmatrix}B \begin{pmatrix} -3 \\ -6 \end{pmatrix}C \begin{pmatrix} 2 \\ 1 \end{pmatrix}D \begin{pmatrix} 3 \\ 6 \end{pmatrix}E \begin{pmatrix} 1 \\ 3 \end{pmatrix}A \vec{AC} = \vec{AB} + \vec{BC}B \vec{AB} = -\vec{BA}C magnitude \begin{pmatrix} 3 \\ 4 \end{pmatrix} = 5D 2\begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}E vectors have no direction30. What is \begin{pmatrix} 1 \\ 5 \end{pmatrix} + \begin{pmatrix} 3 \\ 2 \end{pmatrix}?
Easy A \begin{pmatrix} 4 \\ 7 \end{pmatrix}B \begin{pmatrix} 3 \\ 10 \end{pmatrix}C \begin{pmatrix} 4 \\ 3 \end{pmatrix}D \begin{pmatrix} 2 \\ 3 \end{pmatrix}31. What is 4 × \begin{pmatrix} 0 \\ 2 \end{pmatrix}?
Easy A \begin{pmatrix} 0 \\ 8 \end{pmatrix}B \begin{pmatrix} 4 \\ 6 \end{pmatrix}C \begin{pmatrix} 0 \\ 6 \end{pmatrix}D \begin{pmatrix} 4 \\ 8 \end{pmatrix}32. \vec{OA} = \mathbf{a} and \vec{OB} = \mathbf{b}. M is the midpoint of AB. Find \vec{OM} in terms of \mathbf{a} and \mathbf{b}.
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