Answer to Question #87342 in Mechanics | Relativity for Igwe Juliet Nneka

Question

1. What are the properties of two vectors a and b such that
(a) a+b = c
(b) a+b = a-b
(c) a+b = c and a^2 + b^2 = c^2
2. Given two vectors a = 3i + 2j, b = -1 + 7j. Find a vector c such that a+b+c = 0.
3. Consider two vectors, one of magnitude 3 units and the other of magnitude 4 units. Show how the displacement vectors may be combined to obtain a resultant displacement of magnitude.
(a) 7 units
(b) 1 unit
(c) 5 units

Answer 1

1.

(a) Vectors a and b lie in the same plane

(b) Vectors a and b are perpendicular to each other.

(с) Vectors a and b are perpendicular to each other.

2.For

$$\vec{a}+\vec{b}+\vec{c}=0$$

then

$$\vec{a}+\vec{b}=-\vec{c}$$ $$\vec{a}+\vec{b}=3i+2j-i+7j=2i+9j$$

and

$$\vec{c}=(-1)(\vec{a}+\vec{b})=-2j-9j$$

3.For this problem use it

$$|\vec{a}+\vec{b}|=\sqrt{(\vec{a}+\vec{b})^2}=a^2+b^2+2\vec{a}\cdot\vec{b}$$

where dot product

$$\vec{a}\cdot\vec{b}=|a| \cdot |b| \cdot cos(\phi)$$

(a) when angle between the two vectors is 0

$$|\vec{a}+\vec{b}|=\sqrt{(\vec{a}+\vec{b})^2}=\sqrt{a^2+b^2+2ab }=a+b$$

$$a+b=4+3=7$$

(b) when angle between the two vectors is 180

$$|\vec{a}+\vec{b}|=\sqrt{(\vec{a}+\vec{b})^2}=\sqrt{a^2+b^2-2ab }=a-b$$

$$a-b=4-3=1$$

(c) when angle between the two vectors is 90

$$|\vec{a}+\vec{b}|=\sqrt{(\vec{a}+\vec{b})^2}=\sqrt{a^2+b^2}$$

$$\sqrt{a^2+b^2}=\sqrt{3^2+4^2}=5$$