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(5) Razona la necesidad de la existencia de procesos de apoptosis en las células de un organismo pluricelular.

( $5$ ) Razona la necesidad de la existencia de procesos de apoptosis en las células de un organismo pluricelular.

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Find the magnitude of a three-dimensional vector

Full solution

Q. (

5

) Razona la necesidad de la existencia de procesos de apoptosis en las células de un organismo pluricelular.

Identify Points and Vectors: Identify points and vectors. $\newline$ P $(1,2,3)$ , $O$ is the origin, $Q$ is on the positive $Z$ axis. $\newline$ Vector $OP = P - O = (1,2,3)$ . $\newline$ Vector $OQ = Q - O = (0,0,z)$ .
Calculate Dot Product: Calculate the dot product of $OP$ and $OQ$ . $OP \cdot OQ = (1 \cdot 0) + (2 \cdot 0) + (3 \cdot z) = 3z$ .
Find Magnitude of OP: Find the magnitude of vector OP. $\newline$ $|OP| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{14}.$
Find Magnitude of OQ: Find the magnitude of vector OQ. $\newline$ |OQ| = \sqrt{\(0\)^\(2\) + \(0\)^\(2\) + z^\(2\)} = z.\(\newline
Use Dot Product for Cosine: Use the dot product and magnitudes to find the cosine of angle QOP. $\cos(\theta) = \frac{OP \cdot OQ}{|OP| \cdot |OQ|} = \frac{3z}{\sqrt{14} \cdot z}$ .
Simplify Cosine of QOP: Simplify the cosine of angle QOP. $\cos(\theta) = \frac{3}{\sqrt{14}}$ .
Calculate Angle QOP: Calculate the angle QOP using the inverse cosine function. $\theta = \cos^{-1}\left(\frac{3}{\sqrt{14}}\right)$ .
Use Calculator for Angle: Use a calculator to find the angle. $\theta \approx 70.53^\circ$ (This is the wrong value, the correct value should be closer to $36.7^\circ$ ).

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