Math 10 Chapter 1 Lesson 3: Product of a vector with a number
1. Summary of theory
1.1. Define the product of a vector and a number
The product of the vector \(\vec{a}\) with the real number k is a vector, denoted by \(k\vec{a}\), defined as follows:

If \(k\geq 0\) then the vector \(k\vec{a}\) is in the same direction as the vector \(\vec{a}\).

If \(k<0\) then the vector \(k\vec{a}\) is in the opposite direction to the vector \(\vec{a}\).

The length of the vector \(k\vec{a}\) is equal to \(k.\vec{a}\).
1.2. Properties of vector multiplication with numbers
For any two vectors \(\vec{a}\), \(\vec{b}\) and any real numbers k, l we have:
 \(k\left( {l\overrightarrow a } \right) = \left( {kl} \right)\overrightarrow a\)
 \(\left( {k + l} \right)\overrightarrow a = k\overrightarrow a + l\overrightarrow a \)
 \(k\left( {\overrightarrow a + \overrightarrow b } \right) = k\overrightarrow a + k\overrightarrow b ;\,\,\,k\left( {\overrightarrow a – \overrightarrow b } \right ) = k\overrightarrow a – k\overrightarrow b \)
 \(k\overrightarrow a = \overrightarrow 0 \) if and only if \(k = 0\) or \(\vec{a} =\vec{0}\).
1.3. Condition for two vectors to have the same direction
The vector \(\vec{b}\) is in the same direction as the vector \(\vec{a}\neq \vec{0}\) if and only if there exists a number k such that \(\vec{b}=k\ vec{a}\)
Application to three collinear points:
A necessary and sufficient condition for the three points A, B, and C to be collinear is the number k such that \(\vec{AB}=k\vec{AC}\)
1.4. Express a vector over two vectors that are not in the same direction
Based on the above figure, we have the following theorem:
Given two unequal vectors \(\vec{a}\) and \(\vec{b}\). Then every vector \(\vec{x}\) can be uniquely displayed over two vectors \(\vec{a}\) and \(\vec{b}\), that is, there are pairs of numbers only m and n such that:
\(\vec{x}=m\vec{a}+n\vec{b}\)
2. Illustrated exercise
Question 1: Given vector \(\overrightarrow a \ne \overrightarrow 0 \) . Determines the length and direction of the vector \(\overrightarrow a + \overrightarrow a\)
Solution guide
We have:
\(\overrightarrow a + \overrightarrow a = 2\overrightarrow a \)
The length of the vector \(\overrightarrow a + \overrightarrow a\) is twice the length of the vector \(\overrightarrow a\)
The direction of the vector \(\overrightarrow a + \overrightarrow a\) is the same as the vector \(\overrightarrow a\) (because 2 > 0).
Verse 2: Find the reciprocal of vectors: \(k\overrightarrow a ;\,\,3\overrightarrow a – 4\overrightarrow b \)
Solution guide
The reciprocal of the vectors \(k\overrightarrow a \) is the vector \(k\overrightarrow a \)
The reciprocal vector of vectors \(3\overrightarrow a – 4\overrightarrow b \) is the vector (\(3\overrightarrow a – 4\overrightarrow b \)) or \( – 3\overrightarrow a + 4\overrightarrow b \ ).
3. Practice
3.1. Essay exercises
Question 1: Let OAB isosceles triangle with \(OA = OB = a\) . Calculate the length of the vectors \(\overrightarrow {AO} + \overrightarrow {OB} + \overrightarrow {AB} ;\) \(3\overrightarrow {OA} – 2\overrightarrow {OB} \)
Verse 2: Prove that for any quadrilateral ABCD, we always have the relation: \(\overrightarrow {BA} – \overrightarrow {BC} = \overrightarrow {DA} – \overrightarrow {DC} \) .
Question 3: Let ABC be a triangle. M is the midpoint of BC. Prove that: \(\overrightarrow {AM} = \frac{1}{2}\overrightarrow {AB} + \frac{1}{2}\overrightarrow {AC} \)
3.2. Multiple choice exercises
Question 1: Let ABC be a triangle. Let M be a point on side BC such that MB = 2MC. Then which of the following statements is true?
A. \(\overrightarrow {AM} = \frac{1}{3}\overrightarrow {AB} + \frac{2}{3}\overrightarrow {AC} \)
B. \(\overrightarrow {AM} = \frac{2}{3}\overrightarrow {AB} + \frac{1}{3}\overrightarrow {AC} \)
C. \(\overrightarrow {AM} = \overrightarrow {AB} + \overrightarrow {AC} \)
D. \(\overrightarrow {AM} = \frac{2}{3}\overrightarrow {AB} – \frac{1}{3}\overrightarrow {AC} \)
Verse 2: Given quadrilateral ABCD; X is the centroid of triangle BCD, G is the centroid of quadrilateral ABCD. Which of the following assertion true?
A. \(\overrightarrow {GA} + \overrightarrow {GX} = \overrightarrow 0 \)
B. \(\overrightarrow {GA} +3 \overrightarrow {GX} = \overrightarrow 0 \)
C. \(\overrightarrow {GB} + \overrightarrow {GX} = \overrightarrow 0 \)
D. \(\overrightarrow {GC} + \overrightarrow {GX} = \overrightarrow 0 \)
Question 3: Triangle ABC has the centroid G, the lengths of the sides BC, CA, AB are a, b, c respectively. Then ABC is an equilateral triangle if any of the following conditions exist?
A. \(a\overrightarrow {GA} + b\overrightarrow {GB} + c\overrightarrow {GC} = \overrightarrow 0 \)
B. \(a\overrightarrow {GA} + b\overrightarrow {GB} – c\overrightarrow {GC} = \overrightarrow 0 \)
C. \(a\overrightarrow {GA} – b\overrightarrow {GB} + c\overrightarrow {GC} = \overrightarrow 0 \)
D. \( – a\overrightarrow {GA} + b\overrightarrow {GB} + c\overrightarrow {GC} = \overrightarrow 0 \)
Question 4: Find the false statement:
A. Two vectors in the same direction as another third vector \(\vec{0}\) have the same direction
B. Two vectors in the same direction as another third vector \(\vec{0}\) have the same direction
C. Three vectors \(\vec{a},\vec{b},\vec{c}\) other than \(\vec{0}\) with the same direction, at least two vectors have the same direction.
D. For \(\vec{a}\) and \(\vec{b}\) to be equal, then \(\vec{a}=\vec{b}\)
Question 5: Given the vector \(\overrightarrow a ,\overrightarrow b \) and the real numbers m, n, k. Which of the following assertion is true?
A. From the equality \(m\vec {a} = n\vec {b} \) deduce m = n
B. From the equality \(k\vec {a} = k\vec {kb} \) always deduce \(\vec a = \vec b \)
C. From the equality \(k\vec {a} = k\vec {kb} \) always deduce k = 0
D. From the equality \(m\vec {a} = n\vec {b} \) and \(\vec a \ne \vec 0\) deduce m = n
Question 6: Given three distinct points A, B, C such that \(\overrightarrow {AB} = k\overrightarrow {AC} \). Given that B lies between A and C. Which of the following conditions does the value of k satisfy?
A.k < 0
B.k = 1
C. 0 < k < 1
D.k > 1
Through this lesson, you should know the following:
 Defines the product of a vector with a number.
 Identify two vectors in the same direction.
 How to represent a vector in terms of two vectors that are not in the same direction.
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