STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

1.28 Direction Cosines

The cosines of the angles (a, b, g ) that a vector makes with each of the cartesian axes are called direction cosines. They are used for more practical solutions in some calculations related to strength.

Direction Cosines:

Now we will examine various examples to better establish our knowledge about vectors in our minds….>>

1. Vectors and Vector Operations

(1.17)

Figure 1.2

STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

Examples 1.1

Solution:

y

x

θ

a-)

b-)

1. Vectors and Vector Operations

STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

Example 1.2

Find the components and projections of the 800 N force F in the a-) x and y axes,

b-) a-b directions.

Solution:

a-)

Components in x and y directions;

Note: Since x-y are perpendicular to each other, the components also equal the projections.

Components on the a and b axes;

b-)

1. Vectors and Vector Operations

Projectios on the a and b axes;

STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

Example 1.3:

A(0,1,2) , B(2,0,0)

Solution:

Questions we found answers to in this example:

1- How do we find the vector expression when the magnitude of a vector and the coordinates of 2 points on the vector line are known?

2- How do we find a position vector whose starting and ending points are known and a unit vector in the same direction?

• According to Equation 1.4, a unit vector is equal to a position vector in its own direction divided by its magnitude:

• According to Equation 1.5, the vector expression of a vector is found by multiplying the magnitude of this vector by the unit vector in the same direction:

Suppose you inherited Zulfikar from Ali. If you don't have Ali's heart, what's the use of Zulfikar? (Rumi)

1. Vectors and Vector Operations

STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

Example 1.4

• The vector product (multiplication) of two vectors gives another vector perpendicular to the plane they are in (in the direction of the plane normal). Its direction is found according to the right-hand rule.

1. Vectors and Vector Operations

STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

Questions we found answers to in this example:

1-How do we find the vector expression of a vector perpendicular to a diagonal (inclined) plane when its magnitude is known?

2-How do we find the unit vector in the normal direction of a diagonal plane?

1. Vectors and Vector Operations

A(0,1,2), B(2,0,0), C(2,1.5,0)

STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

1. Vectors and Vector Operations

If the magnitude of the V vector in the system shown is 200 and the magnitude of the P vector is 600, find the vector expressions of V and P.

Question 1.1

STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

Since

(Industrial Engineering Static 1st midterm-2007)

Question 1.2

1. Vectors and Vector Operations

Calculate the operation

;

;

STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

1. Vectors and Vector Operations

Question 1.3 (*)

STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

The vectors shown in the figure are:

and

It is given as

a-) Calculate the value of the angle 𝜃 between these vectors.

b) Calculate the magnitude of the sum of these vectors.

Answers: a-) 70.55^o b-) 10.05

1. Vectors and Vector Operations

Question 1.4 (*)

Accordingly;

STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

1. Vectors and Vector Operations

Question 1.5 (*)

Answer:

STATICS - Lecture Notes / Mehmet Zor

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26 September, 2025

Question 1.6

If F₁=10kN (applied perpendicular to the middle of the DCBF plane); M₁= 20kNm (applied to the EFD plane), express the F₁ and M₁ values ​​as vectors.

Answers:

1. Vectors and Vector Operations