Scalar product: math exercises in 1st grade corrected in PDF.

Corrected exercises of maths 1st on the scalar product in the plane.

You will find in these exercises on the scalar product the following notions:

definition of the scalar product;
bilinearity of the scalar product;
symmetry of the scalar product;
identity of the parallelogram;
scalar product and orthogonal vectors;
Cartesian and parametric equations.

The scalar product in the plane is a tool used to measure the angle between two vectors. It is defined as the product of the norm (or length) of two vectors and the cosine of the angle between them.

Bilinearity is a property of a function that allows the function to be decomposed into two linear products.

Exercise 1:

Consider the square ABCD with center O and side 8.

Calculate the following scalar products.

$a)\vec{AB}.\vec{AO}$ $b)\vec{OB}.\vec{OD}$

$c)\vec{AB}.\vec{AD}$ $d)\vec{BO}.\vec{BC}$

Exercise 2:

Consider the vectors $\vec{u}$ and $\vec{v}$ such that $\%7C\vec{u}\%7C=2$ , $\%7C\vec{v}\%7C=3$ and $\widehat{(\vec{u},,\vec{v})}=60^{\circ}$ .

Calculate their scalar product.

Exercise 3:

Determine a value in degrees of the angle between the vectors $\vec{u}$ and $\vec{v}$ such that $\%7C\vec{u}\%7C=6$ , $\%7C\vec{v}\%7C=2$ and $\vec{u}.\vec{v},=-6$ .

Exercise 4:

Let be the vectors $\vec{u},(-2;3,)$ and $\vec{v}(-1;-5)$ .

Calculate:

$a)\vec{u}.\vec{v}$ $b)(4\vec{u}).\vec{v}$ $c)(\vec{u}-\vec{v}).(\vec{u}+\vec{v})$

Exercise 5:

We give the points A(-3;-2) and B(1;3) and the vector $\vec{u}(,-5;4,)$ .

Show that $\vec{AB}$ and $\vec{u}$ are orthogonal.

Exercise 6:

A,B,C and D being any points of the plane, show the following equalities.

$a)\vec{AB}.\vec{CD}=\vec{BA}.\vec{DC}$ .

$b)\vec{AB}.\vec{AC}+\vec{AB}.\vec{EC}=\vec{AB}.\vec{ED}$

$c)\vec{AB}.\vec{AC}=\vec{AB}^2-\vec{BA}.\vec{BC}$

Exercise 7:

We give the points C and D such that CD = 10 and H the middle of the segment [CD].

Determine the set of points M of the plane verifying $\vec{MC}.\vec{MD}=-9$ .

Exercise 8:

In a rectangle ABCD of length 8 and width 4, we place the points E, F and G such that :

$\vec{AE}=\frac{1}{4}\vec{AD};\vec{AG}=\frac{1}{8}\vec{AB};\vec{CF}=\frac{1}{4}\vec{CB}$ .

1. In the reference frame (A ; G,E), give the coordinates of all the points of the figure.

2. Calculate the scalar product $\vec{EF}.\vec{DG}$ .

3. What can we deduce from this?

Exercise 9:

ABCD is a rectangle of center F and E is the symmetrical of point F with respect to the line
(BC). Calculate the following scalar products.

$a)\vec{BA}.\vec{BF}\,;b)\vec{CF}.\vec{CD}\,;c)\vec{AF}.\vec{AB}\,;d)\vec{AB}.\vec{BE}$

Exercise 10:

Let be the vectors $\vec{u}(2;1)$ , $\vec{v}(-3;-1)$ and $\vec{w}(1;4)$ .

Calculate the following scalar products.

$a)\vec{u}.\vec{v}\\b)\vec{w}.\vec{v}\\c)\vec{u}.(\vec{v}+\vec{w})\\d)(-2\vec{u}).\vec{v}+3(\vec{v}.\vec{w})$

Exercise 11:

We give the vectors $\vec{u}(-3;4)$ and $\vec{v}(-8;-6)$ .

Show that these vectors are orthogonal.

Exercise 12:

Give a direction vector for each of the following lines and deduce that they are perpendicular.
a) For the lines d1 and d2 of Cartesian equations 2x-3y+4=0 and 3x+2y-1= 0.
b) For the lines d1and d2 of Cartesian equations x-y+3=0 and 2x+2y-1=0.
c) For the lines d1 and d2 of equations y = -3x + 1 and -x+3y-1=0.

Exercise 13:

Let be the vectors $\vec{u}(-2;3)$ , $\vec{v}(-1;-5)$ .

Calculate:

$a)\vec{u}.\vec{v}\\b)(4\vec{u}).\vec{v}\\c)(\vec{u}-\vec{v}).(\vec{u}+\vec{v})$

Exercise 14:

Let be the vectors $\vec{u}(-3;4)$ , $\vec{v}(-8;-6)$ .

Show that these vectors are orthogonal.

2. We give the points A(-3;-2) and B(1;3) and the vector $\vec{u}(-5;4)$ .

Show that $\vec{AB}$ and $\vec{u}$ are orthogonal.

Exercise 15:

Consider points A, B and C such that AB = 3, AC = 4 and $\widehat{BAC}$ = 120°.

Determine the length BC.
2. Consider the points M, N and P such that MN = 5, NP = 7 and MNP = 61°.

Determine the MP length.
3. Let EFG be a triangle such that EF = 7, FG = 6 and EG = 11.
Determine the value in degrees and rounded to 0.1° of the angle $\widehat{EFG}$ .
4. Let EDF be a triangle such that EF = 5, DF = 8 and ED = 9.
Determine the value in degrees and rounded to 0.1° of the angle $\widehat{EDF}$ .

Exercise 16:

be the vectors $\vec{u}$ and $\vec{v}$ orthogonal and such that $\,\%7C\vec{u}\,\,\%7C=a$ and $\,\%7C\vec{v}\,\,\%7C=b$ .

Express in terms of a and b the following scalar products.

$a)\vec{u}.(\vec{u}+\vec{v})\\b)(2\vec{u}-3\vec{v}).\vec{v}\\c)(\vec{u}+\vec{v})^2$

Exercise 17:

Let the vectors $\vec{u}$ ; $\vec{v}$ and $\vec{w}$ be such that : $\,\%7C\,\vec{u\,}\,\%7C=\%7C\vec{w\,}\,\,\%7C=a$ and $\vec{v}=3\vec{u}$ .

The vectors $\vec{u}$ and $\vec{w}$ are orthogonal.

Express in terms of a the following scalar products.

$a)\vec{u}.\vec{v}\\b)\vec{v}.(\vec{u}+\vec{w})\\c)(\vec{u}+\vec{v})^2\\d)(\vec{v}+\vec{w}).(\vec{u}+\vec{w})$

Exercise 18:

A, B, C and D being any points of the plane, show the following equalities.
$a)\vec{AB}.\vec{CD}\,=\,BA.\vec{DC}\\b)\vec{AB}.\vec{CD}+\vec{AB}.\vec{EC}=\vec{AB}.\vec{ED}\\\,c)\vec{AB}\,.\vec{AC}=\vec{AB}\,-\vec{BA\,}-\vec{BC}$

Exercise 19:

1. we give the points A and B such that AB = 12 and I the middle of the segment [AB].

Determine the set of points M of the plane verifying $\vec{MA}\,.\vec{MB}\,=\,4$ .

2. points C and D are given such that CD = 10 and H is the middle of the segment [CD]. Determine the set of points M of the plane verifying $\vec{MC}.\vec{MD\,}=\,-\,9$ .

Exercise 20:

Consider a right-angled trapezoid ABCD such that the diagonal [AC] is perpendicular to side [BC]. By calculating the scalar product $\vec{AB}.\vec{\,AC}$ in two ways, demonstrate
that $AC^2\,=\,AB\times \,CD$ .

Exercise 21:

We consider two squares ABCD and BEFG arranged as shown in the figure
below such that AB = 1 and BE = a.

A. With coordinates
1. In the reference frame (A ; B, D), give the coordinates of all the points of the figure.
2. Show that the lines (AG) and (CE) are perpendicular.

B. Without contact information
1. Develop the scalar product $(\vec{AB}\,+\,\vec{BG}).\,(\vec{CB}\,+\,\vec{BE})$ .
2. Deduce that $\vec{AG}\,.\vec{CE}\,=\,0$ and that the lines (AG) and (CE) are perpendicular.

Exercise 22:

ABCD is a square of side a and AEFG is a square of side b with D, A and G aligned, as well as B, A and E as in the figure below.

The point I is the middle of the segment [DE].

A. Without contact information
1. Justify that AD + AE = 2Al.
2. Develop the scalar product (AD + AE) . (BA + AG).
3. Deduce that the lines (AI) and (BG) are perpendicular.

B. With coordinates
1. In the reference frame (A ; B, D) give the coordinates of points A, I, B and G.
2. Deduce that the lines (AI) and (BG) are perpendicular.

Exercise 23:

Consider a square ABCD of side 1 and a point M on the segment [BD]. We construct the orthogonal projects H and K of point M on
the sides [AB] and [AD].

1. We want to show that the lines (CK) and (DH) are perpendicular by two methods:
a) We will use the reference frame (A; B, D) and we will note (x;y) the coordinates of the point M.
b) We will calculate the scalar product: $\vec{CK}.\vec{DH}$ by decomposing the vectors using the Chasles relation.
2. Show that the lengths CK and DH are equal:
a) with coordinates.
b) without coordinates.

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Scalar product: math exercises in 1st grade corrected in PDF.

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