Question

7 Find a unit vector parallel to the resultant vector A1=2i+4j−5k, A2=1+2j+3k
8 Given the scalar defined by ϕ(x,y,z)=3x2z−xy2+5,find ϕ at the points (-1,-2,-3)
9 The following forces act on a particle P:F1=2i+3j−5k, F2=−5i+j+3k,F3=i−2j+4k,F4=4i−3j−2k, Find the magnitude of the resultant
10 If a and b are non-collinear vectors and A=(x+y)a+(2x+y+1)b

Answer 1

Answer on Question #59202 – Math – Analytic Geometry

Question

7 Find a unit vector parallel to the resultant vector $A1 = 2i + 4j - 5k$, $A2 = i + 2j + 3k$

Solution

The resultant vector is

$$\bar{A} = \overline{A_1} + \overline{A_2} = (2\bar{i} + 4\bar{j} - 5\bar{k}) + (\bar{i} + 2\bar{j} + 3\bar{k}) = (2 + 1)\bar{i} + (4 + 2)\bar{j} + (-5 + 3)\bar{k} = 3\bar{i} + 6\bar{j} - 2\bar{k}$$

The magnitude of the resultant vector is

$$|\bar{A}| = \sqrt{(3)^2 + (6)^2 + (-2)^2} = 7$$

A unit vector parallel to the resultant vector is

$$e_A = \pm \frac{\bar{A}}{|\bar{A}|} = \pm \frac{3\bar{i} + 6\bar{j} - 2\bar{k}}{7}.$$

**Answer:** $\pm \frac{3\bar{i} + 6\bar{j} - 2\bar{k}}{7}$.

Question

8 Given the scalar defined by $\varphi(x,y,z) = 3x^2z - xy^2 + 5$, find $\varphi$ at the points $(-1, -2, -3)$

Solution

$$\phi(x, y, z) = 3x^2z - xy^2 + 5$$ $$\phi(-1, -2, -3) = 3(-1)^2(-3) - (-1)(-2)^2 + 5 = -9 + 4 + 5 = 0.$$

**Answer:** 0.

Question

9 The following forces act on a particle $P$: $F1 = 2i + 3j - 5k$, $F2 = -5i + j + 3k$, $F3 = i - 2j + 4k$, $F4 = 4i - 3j - 2k$, Find the magnitude of the resultant

Solution

The resultant force is

$$\begin{aligned} \bar{F} &= \bar{F_1} + \bar{F_2} + \bar{F_3} + \bar{F_4} = (2\bar{i} + 3\bar{j} - 5\bar{k}) + (-5\bar{i} + \bar{j} + 3\bar{k}) + (\bar{i} - 2\bar{j} + 4\bar{k}) + (4\bar{i} - 3\bar{j} - 2\bar{k}) \\ &= (2 - 5 + 1 + 4)\bar{i} + (3 + 1 - 2 - 3)\bar{j} + (-5 + 3 + 4 - 2)\bar{k} = 2\bar{i} - \bar{j} - 0\bar{k} \end{aligned}$$

The magnitude of the resultant force is

$$|\bar{F}| = \sqrt{(2)^2 + (-1)^2 + (0)^2} = \sqrt{5}.$$

**Answer:** $\sqrt{5}$.

Question

10 If $a$ and $b$ are non-collinear vectors and $A = (x+y)a + (2x+y+1)b$

Answer: the statement of question is not complete. What should be done there?

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