Scalar product : 11th grade math worksheets with answers in PDF

11th grade math worksheets

Scalar product in the plane with 11th grade math worksheets online to progress to high school. Apply the properties of the scalar product and demonstrate that vectors are orthogonal or collinear. Use the Chasles relation on vectors.

Exercise 1:
Let $\,\vec{AB}$ and $\,\vec{AC}$ be two vectors and $\,k\in\mathbb{Z}$ .
Calculate $\,\vec{AB}.\vec{AC}$ under the following conditions:
a. AB=3 , AC=5 and $\,(\vec{AB}.\vec{AC})=-\frac{\pi}{6}+2k\pi$ .
b. AB=1 , AC=4 and $\,(\vec{AB}.\vec{AC})=-\frac{8\pi}{3}+2k\pi$ .
c. AB=4 , AC=7 and $\,(\vec{AB}.\vec{AC})=-\frac{\pi}{4}+2k\pi$ .
d. AB=2 , AC=2 and $\,(\vec{AB}.\vec{AC})=-\frac{5\pi}{3}+2k\pi$ .

Exercise #2:
Calculate $\,\vec{AC}.\vec{AB}\,;\,\vec{CA}.\vec{BA}\,;\,\vec{BA}.\vec{AC}\,\,;$ knowing that :
a. $\,\vec{AB}.\vec{AC}=-3$
b. $\,\vec{AB}.\vec{AC}=2$

Exercise #3:
MNPQ is a rhombus of center O such that MP=8 and NQ=6.
Calculate the following scalar products:
a. $\,\vec{MO}.\vec{MN}\,;\,\vec{PQ}.\vec{NQ}\,;\,\vec{PM}.\vec{NP}\,\,;$ .
b. $\,\vec{MQ}.\vec{NP}\,;\,\vec{MN}.\vec{PQ}\,;\,\vec{OM}.\vec{NM}\,\,;$

Exercise #4:
Let ABCD be a square and I a point of [AB].
Let H be the orthogonal project of A onto [ID].
By expressing in two different ways $\,\vec{IA}.\vec{ID}$ , show that :
$\,\vec{IA}.\vec{ID}=AI^2$

Exercise #5:
Let ABC be an equilateral triangle of side 1.
Let H be the orthogonal project of A onto (BC).
Calculate $\,\vec{BA}.\vec{AC}$ and $\,\vec{AB}.\vec{AH}$ using orthogonal projections.

Exercise 6 – Scalar product in a square

Let be a square ABCD. We construct a rectangle APQR such that :

– P and R are on the sides [AB] and [AD] of the square ;
– AP = DR.

The purpose of the problem is to show that the lines (CQ) and (PR) are perpendicular.1. Justify that: $\vec{CQ}.\vec{PR}=\vec{CQ}.(\vec{AR}-\vec{AP})$

.2. Deduce that the lines (CQ) and (PR) are perpendicular.

Exercise 7 – Algebraic properties
We have $\,\%7C\,\vec{u}\,\,\%7C=2$

and $\,\%7C\,\vec{v}\,\,\%7C=3$

and $\vec{u}$ . $\vec{v}$ = -1.
1) Calculate $(\vec{u}+\vec{v})^2$

and $\,\%7C\,(\vec{u}\,-\vec{v})^2\,\%7C$

.
2) Calculate ( $\vec{u}$ + $\vec{v}$ ) . (2 $\vec{u}$ -3 $\vec{v}$ ).Exercise 8 – Scalar product and any point
Let A and B be two distinct points of the plane and I the middle of the segment [AB].
Prove that whatever the point M of the plane, we have the equality :
$MA^2-MB^2=(\vec{MA}+\vec{MB}).\vec{BA}=2\vec{MI}.\vec{BA}$

.Exercise 9 – Vectors in the plane
Let be the parallelogram ABCD such that :
E is the middle of [AD]
$\vec{AF}=\frac{2}{3}\vec{AB}$

K is the last vertex of the parallelogram EAFK
M the middle of [BE]
$\vec{AG}=\frac{1}{3}\vec{AB}$

$\vec{GB}=2\vec{GF}$

$\vec{GC}=2\vec{GK}$

Show that vector $\vec{GK}=2\vec{GM}$ .

Exercise 10 – Orthogonal Project
ABC is a right-angled triangle in A .
H is the orthogonal project of A on (BC) .
I and J are the respective middles of [AB] and [AC].

Show that (HI) and (HJ) are perpendicular.

Exercise 11 – Calculation of scalar products in a parallelogram

ABCD is a parallelogram with AB = 4, AD = 5 and AC = 7.

1.Calculate $\vec{AB}.\vec{AD}$ .

2. Deduct BD.

Exercise 12 – Calculation of scalar products in a square
MNPQ is a square with MN = 6. I is the center of the square.

Calculate the following scalar products:

1. $\vec{MN}.\vec{QP}.$

2. $\vec{MN}.\vec{PN}.$

3. $\vec{IN}.\vec{IP}.$

4. $\vec{QI}.\vec{NI}.$

Exercise 13 – Determining if the triangle is right-angled

ABC is a triangle in which AB = 2 and AC = 3.

In addition $\vec{AB}.\vec{AC}=4$

Is this triangle rectangular? If yes, specify in which summit.

Exercise 14 – Equilateral triangle
ABC is an equilateral triangle of side 5 cm. I is the middle of [BC].

Calculate the following scalar products:

1. $\vec{BA}.\vec{BC}$ .

2. $\vec{CA}.\vec{CI}.$

3. $(\vec{AB}-\vec{AC}).\vec{AI}.$

Exercise 15 – Coordinates of the barycentre

In an orthonormal reference frame $(O;\vec{i},\vec{j})$
we consider the following points: A (2; 1), B (7; 2) and C (3; 4).

All of the following questions are independent and unrelated.

1. Compute the coordinates of the barycenter G of (A; 3), (B; 2) and (C; – 4).

2. Determine a Cartesian equation of the perpendicular bisector of [BC].

3. Calculate $\vec{CB}.\vec{CA}$ .

4. Is the angle $\widehat{C}$ right?

Exercise 16 – Cosine
Let ABC be a triangle.
Calculate $\vec{AB}.\vec{AC}$ and in each of the following cases:
1. AB= 6cm; AC= 5 cm and $\widehat{BAC}=60^{\circ}$ .
2. AB= 7 cm; AC=4cm and $\widehat{BAC}=120^{\circ}$ .

Exercise 17 – Orthogonal vectors
$\vec{u}$ and $\vec{v}$ are two vectors of the same norm .
Show that the vectors $\vec{u}+\vec{v}$ and $\vec{u}-\vec{v}$ are orthogonal.

Exercise 18 – Equilateral triangle
ABC is an equilateral triangle of side .
H is the orthogonal project of A onto (BC) and O is the center of the circumscribed circle of ABC.
Express in terms of , the following scalar products:
$\vec{AB}.\vec{AC}\,;\,\vec{AC}.\vec{CB}\,;\,\vec{AB}.\vec{AH}\,;\,\vec{AH}.\vec{BC}\,;\,\vec{OA}.\vec{OB}\,$ .

Exercise 19 – Calculations with scalar products
Knowing that the vectors $\vec{u}$ and $\vec{v}$ are such that $\,\%7C\,\vec{u}\,\,\%7C=3$ , $\,\%7C\,\vec{v}\,\,\%7C=7$ and $\vec{u}.\vec{v}\,=13$ .
Calculate the following scalar products:
1. $\vec{u}.\,(\vec{u}+3\vec{v}\,\,)$ .
2. $\,(\vec{u}-2\vec{v}\,\,)\,^2$ .

Exercise 20 – Condition on points

Under which condition on points A, B and C do we have :

$(\vec{AB}+\vec{AB})^2=(AB+AC)^2$

Exercise 21 – Determine a set of points in the plane

Consider a segment [AB] such that AB = 1 dm.

Determine the set of points M in the plane such that :

1. $\vec{MA}.\vec{MB}=1.$

Exercise 22 – Finding a set of points
[AB] is a segment of middle I and AB = 2 cm.
1. Show that for any point M in the plane :
$MA^2-MB^2=2\vec{IM}.\vec{AB}$
2. Find and represent the set of points M in the plane such that : $MA^2\,-MB^2\,=\,14.$

Exercise 23 – Vector Equalities of the Parallelogram
Prove that :
1. $\,\%7C\,\vec{u}+\vec{v}\,\,\%7C^2-\,\%7C\,\vec{u}-\vec{v}\,\,\%7C^2=4\vec{u}.\vec{v}$ .
2. $\,\%7C\,\vec{u}+\vec{v}\,\,\%7C^2+\,\%7C\,\vec{u}-\vec{v}\,\,\%7C^2=2(\,\%7C\vec{u}\,\,\%7C^2+\,\%7C\,\vec{v}\,\,\%7C^2)$ .
3. What is the link with the rhombus, the parallelogram?
4. Show that :
$(\vec{u}+\vec{v}).(\vec{u}-\vec{v})=\,\%7C\,\vec{u}\,\,\%7C^2-\,\%7C\,\vec{v}\,\,\%7C^2$
5. Deduce that a parallelogram has its diagonals perpendicular if and only if its sides are equal.

Exercise 24 – Equation of a circle and its tangent

In an orthonormal reference frame $(O;\vec{i},\vec{j})$ , we give a point $\Omega\,(2;-3)$ .

1. Determine the equation of the circle (C) with center $\Omega$ and radius R = 5.

2. Prove that the point A( – 2 ; 0) is a point of the circle (C).

3. Determine a Cartesian equation of the tangent at A to the circle (C).

Exercise 25 – Median and height of a triangle
MNPQ is a square with MN = 6. I is the center of the square.

Calculate the following scalar products:

1. $\vec{MN}.\vec{QP}.$

2. $\vec{MN}.\vec{PN}.$

3. $\vec{IN}.\vec{IP}.$

4. $\vec{QI}.\vec{NI}.$

Exercise 26 – Distance from a point to a circle
We place ourselves in an orthonormal reference frame $(O;\vec{i},\vec{j})$ .
1. Determine the equation of the circle with center $\Omega\,(5;1)$ tangent to the line (D) of equation :
$x\,+\,y\,-\,4\,=\,0.$
Indication:

we recall that the distance between a point $A(\alpha\,;\beta\,)$ and a line (D) of equation ax + by + c = 0 is
given by the formula :

$d(A,D)=\frac{\,%7C\,a\alpha\,+b\beta\,+c\,\,%7C}{\sqrt{a^2+b^2}}$

Exercise 27 – Scalar product and circle
We place ourselves in an orthonormal reference frame $(O;\vec{i},\vec{j})$ .

Consider whether the following equations are equations of a circle and, if so, specify the center and radius of the circle.

1. $x^2\,+\,y^2\,-\,2x\,-\,6y\,+\,5\,=\,0.$

2. $x^2\,+\,y^2\,-\,x\,-\,3y\,+\,3\,=\,0.$

Exercise 28 – Scalar product in a triangle

ABC is a triangle and I is the middle of [BC].
We give: BC = 4, AI = 3 and $(\vec{IA},\vec{IB})=\frac{\pi}{3}$ .

Calculate:

1. $\vec{AB}.\vec{AC}.$

4. $AB\,et\,AC.$

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