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Sagot :
Let's analyze the given vector components and calculate their properties. We are given the vector components:
[tex]\[ \mathbf{v} = (r, s, t) = (5, -3, 8) \][/tex]
We'll calculate two main properties: the magnitude of the vector and the direction angle using the components [tex]\( r \)[/tex] and [tex]\( s \)[/tex].
### Step 1: Calculate the Magnitude
The magnitude [tex]\( \|\mathbf{v}\| \)[/tex] of a vector in 3-dimensional space [tex]\(\mathbf{v} = (r, s, t)\)[/tex] is given by the formula:
[tex]\[ \|\mathbf{v}\| = \sqrt{r^2 + s^2 + t^2} \][/tex]
Substituting the given values [tex]\( r = 5 \)[/tex], [tex]\( s = -3 \)[/tex], and [tex]\( t = 8 \)[/tex]:
[tex]\[ \|\mathbf{v}\| = \sqrt{5^2 + (-3)^2 + 8^2} = \sqrt{25 + 9 + 64} = \sqrt{98} \][/tex]
Evaluating [tex]\(\sqrt{98}\)[/tex]:
[tex]\[ \sqrt{98} \approx 9.899 \][/tex]
So, the magnitude of the vector is approximately:
[tex]\[ \|\mathbf{v}\| \approx 9.899 \][/tex]
### Step 2: Calculate the Direction Angle (θ)
The direction angle [tex]\( \theta \)[/tex] in the context of a 3D vector can often be understood as the angle between the projection of the vector on the [tex]\( xy \)[/tex]-plane and the positive [tex]\( x \)[/tex]-axis. This can be calculated as:
[tex]\[ \theta = \arctan\left(\frac{s}{r}\right) \][/tex]
Substituting [tex]\( r = 5 \)[/tex] and [tex]\( s = -3 \)[/tex] into the formula:
[tex]\[ \theta = \arctan\left(\frac{-3}{5}\right) \][/tex]
We convert this angle from radians to degrees:
[tex]\[ \theta \approx -30.964^\circ \][/tex]
So the direction angle is approximately:
[tex]\[ \theta \approx -30.964^\circ \][/tex]
### Consider the Given Options and Find the Closest Match
We compare our calculated magnitude and direction angle to the given options:
1. [tex]\(10.8, \theta = 56.3^\circ\)[/tex]
2. [tex]\(18.4, \theta = 119.4^\circ\)[/tex]
3. [tex]\(41.0, \theta = 77.3^\circ\)[/tex]
4. [tex]\(97.6, \theta = 24.2^\circ\)[/tex]
None of the options match our calculated values ([tex]\(\approx 9.899\)[/tex] for magnitude and [tex]\(-30.964^\circ\)[/tex] for direction). Thus, our result is:
[tex]\[ \rho \approx 9.899, \quad \theta \approx -30.964^\circ \][/tex]
[tex]\[ \mathbf{v} = (r, s, t) = (5, -3, 8) \][/tex]
We'll calculate two main properties: the magnitude of the vector and the direction angle using the components [tex]\( r \)[/tex] and [tex]\( s \)[/tex].
### Step 1: Calculate the Magnitude
The magnitude [tex]\( \|\mathbf{v}\| \)[/tex] of a vector in 3-dimensional space [tex]\(\mathbf{v} = (r, s, t)\)[/tex] is given by the formula:
[tex]\[ \|\mathbf{v}\| = \sqrt{r^2 + s^2 + t^2} \][/tex]
Substituting the given values [tex]\( r = 5 \)[/tex], [tex]\( s = -3 \)[/tex], and [tex]\( t = 8 \)[/tex]:
[tex]\[ \|\mathbf{v}\| = \sqrt{5^2 + (-3)^2 + 8^2} = \sqrt{25 + 9 + 64} = \sqrt{98} \][/tex]
Evaluating [tex]\(\sqrt{98}\)[/tex]:
[tex]\[ \sqrt{98} \approx 9.899 \][/tex]
So, the magnitude of the vector is approximately:
[tex]\[ \|\mathbf{v}\| \approx 9.899 \][/tex]
### Step 2: Calculate the Direction Angle (θ)
The direction angle [tex]\( \theta \)[/tex] in the context of a 3D vector can often be understood as the angle between the projection of the vector on the [tex]\( xy \)[/tex]-plane and the positive [tex]\( x \)[/tex]-axis. This can be calculated as:
[tex]\[ \theta = \arctan\left(\frac{s}{r}\right) \][/tex]
Substituting [tex]\( r = 5 \)[/tex] and [tex]\( s = -3 \)[/tex] into the formula:
[tex]\[ \theta = \arctan\left(\frac{-3}{5}\right) \][/tex]
We convert this angle from radians to degrees:
[tex]\[ \theta \approx -30.964^\circ \][/tex]
So the direction angle is approximately:
[tex]\[ \theta \approx -30.964^\circ \][/tex]
### Consider the Given Options and Find the Closest Match
We compare our calculated magnitude and direction angle to the given options:
1. [tex]\(10.8, \theta = 56.3^\circ\)[/tex]
2. [tex]\(18.4, \theta = 119.4^\circ\)[/tex]
3. [tex]\(41.0, \theta = 77.3^\circ\)[/tex]
4. [tex]\(97.6, \theta = 24.2^\circ\)[/tex]
None of the options match our calculated values ([tex]\(\approx 9.899\)[/tex] for magnitude and [tex]\(-30.964^\circ\)[/tex] for direction). Thus, our result is:
[tex]\[ \rho \approx 9.899, \quad \theta \approx -30.964^\circ \][/tex]
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