Question

1. find a unit vector parallel to the resultant vector A1 =2i+4j-5k, A2 = 1+2j+3k
2.given the scaler defind by phi(x,y,z)=3x^2z-xy^2+5. find phi at the points (-1,-2,-3)

Answer 1

Answer on Question #62920 – Math – Analytic Geometry

Question

1. Find a unit vector parallel to the resultant vector

$$\vec {A} _ {1} = 2 \vec {\imath} + 4 \vec {\jmath} - 5 \vec {k}, \vec {A} _ {2} = \vec {\imath} + 2 \vec {\jmath} + 3 \vec {k}.$$

Solution

The resultant vector is

$$\vec {A} = \vec {A} _ {1} + \vec {A} _ {2} = (2 \vec {\imath} + 4 \vec {\jmath} - 5 \vec {k}) + (\vec {\imath} + 2 \vec {\jmath} + 3 \vec {k}) = 3 \vec {\imath} + 6 \vec {\jmath} - 2 \vec {k}.$$

Find the magnitude of the vector $\vec{A}$:

$$\left| \vec {A} \right| = \sqrt {3 ^ {2} + 6 ^ {2} + (- 2) ^ {2}} = \sqrt {4 9} = 7.$$

A unit vector parallel to the resultant vector:

$$\overrightarrow {e _ {A}} = \pm \frac {\vec {A}}{| \vec {A} |} = \pm \frac {3 \vec {\imath} + 6 \vec {\jmath} - 2 \vec {k}}{7}.$$ $$\overrightarrow {e _ {A}} = \left(\frac {3}{7}, \frac {6}{7}, - \frac {2}{7}\right) \text{ or } \overrightarrow {e _ {A}} = \left(- \frac {3}{7}, - \frac {6}{7}, \frac {2}{7}\right).$$

Answer: $\overrightarrow{e_A} = \left(\frac{3}{7},\frac{6}{7}, - \frac{2}{7}\right)$ or $\overrightarrow{e_A} = \left(-\frac{3}{7}, - \frac{6}{7},\frac{2}{7}\right)$.

Question

2. Given the scalar defined by

$$\varphi (x, y, z) = 3 x ^ {2} z - x y ^ {2} + 5,$$

find $\varphi$ at the point $(-1, -2, -3)$.

Solution

$$\varphi (- 1, - 2, - 3) = 3 (- 1) ^ {2} (- 3) - (- 1) (- 2) ^ {2} + 5 = - 9 + 4 + 5 = 0.$$

Answer: $\varphi (-1, - 2, - 3) = 0$

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