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## Sagot :

Given the vectors [tex]\( u = \langle -8, -8 \rangle \)[/tex] and [tex]\( v = \langle -1, 2 \rangle \)[/tex]:

**Step 1: Calculate the dot product of [tex]\( u \)[/tex] and [tex]\( v \)[/tex].**

The dot product [tex]\( u \cdot v \)[/tex] is calculated as follows:

[tex]\[ u \cdot v = (-8)(-1) + (-8)(2) = 8 - 16 = -8 \][/tex]

**Step 2: Calculate the dot product of [tex]\( v \)[/tex] with itself.**

The dot product [tex]\( v \cdot v \)[/tex] is calculated as follows:

[tex]\[ v \cdot v = (-1)^2 + (2)^2 = 1 + 4 = 5 \][/tex]

**Step 3: Calculate the projection of [tex]\( u \)[/tex] onto [tex]\( v \)[/tex].**

The projection formula is given by:

[tex]\[ \operatorname{proj}_v u = \left( \frac{u \cdot v}{v \cdot v} \right) v = \left( \frac{-8}{5} \right) \langle -1, 2 \rangle \][/tex]

So,

[tex]\[ \operatorname{proj}_v u = \left( \frac{-8}{5} \right) \times \langle -1, 2 \rangle = \langle 1.6, -3.2 \rangle \][/tex]

**Step 4: Calculate the orthogonal component.**

The orthogonal component is found by subtracting the projection from [tex]\( u \)[/tex]:

[tex]\[ \text{Orthogonal component} = u - \operatorname{proj}_v u = \langle -8, -8 \rangle - \langle 1.6, -3.2 \rangle = \langle -8 - 1.6, -8 + 3.2 \rangle \][/tex]

So,

[tex]\[ \text{Orthogonal component} = \langle -9.6, -4.8 \rangle \][/tex]

Therefore, [tex]\( u \)[/tex] can be expressed as:

[tex]\[ u = \operatorname{proj}_v u + \text{Orthogonal component} = \langle 1.6, -3.2 \rangle + \langle -9.6, -4.8 \rangle \][/tex]

From the given options:

- a. [tex]\( \langle -0.5, 1 \rangle + \langle -7.5, -9 \rangle \)[/tex]

- b. [tex]\( \langle -1, 2 \rangle + \langle -7, -10 \rangle \)[/tex]

- c. [tex]\( \langle -1, 2 \rangle + \langle -2, -1 \rangle \)[/tex]

- d. [tex]\( \langle 1.6, -3.2 \rangle + \langle -9.6, -4.8 \rangle \)[/tex]

The correct choice is:

**d. [tex]\( \langle 1.6, -3.2 \rangle + \langle -9.6, -4.8 \rangle \)[/tex]**

So, the best answer from the choices provided is:

**D**