# Physics Study Guide/Vectors and scalars

**Vectors** are quantities that are characterized by having both a
numerical **quantity** (called the \"magnitude\" and denoted as \|*v*\|)
and a **direction**. Velocity is an example of a vector; it describes
the time rated change in position with a numerical quantity (meters per
second) as well as indicating the direction of movement.

The definition of a vector is any quantity that adds according to the
parallelogram law (there are some physical quantities that have
magnitude and direction that are not vectors).

**Scalars** are quantities in physics that have **no direction**. Mass
is a scalar; it can describe the quantity of matter with units
(kilograms) but does not describe any direction.

## Multiplying vectors and scalars

-   A **scalar** times a **scalar** gives a **scalar** result.
-   A **vector** scalar-multiplied by a **vector** gives a **scalar**
    result (called the dot-product).
-   A **vector** cross-multiplied by a **vector** gives a **vector**
    result (called the cross-product).
-   A **vector** times a **scalar** gives a **vector** result.

## Frequently Asked Questions about Vectors

##### When are scalar and vector compositions essentially the same?

**Answer:** when multiple vectors are in same direction then we can just
add the magnitudes.so, the scalar and vector composition will be same as
we do not add the directions.

##### What is a \"dot-product\"? (work when force not parallel to displacement)

!A Man walking up a
hill **Answer:**
Let\'s take gravity as our force. If you jump out of an airplane and
fall you will pick up speed. (for simplicity\'s sake, let\'s ignore air
drag). To work out the kinetic energy at any point you simply multiply
the *value* of the force caused by gravity by the *distance* moved in
the direction of the force. For example, a 180 N boy falling a distance
of 10 m will have 1800 J of extra kinetic energy. We say that the man
has had 1800 J of work done on him by the force of gravity.

Notice that energy is *not* a vector. It has a value but no direction.
Gravity and displacement are vectors. They have a value plus a
direction. (In this case, their directions are down and down
respectively) The reason we can get a scalar energy from vectors gravity
and displacement is because, in this case, they happen to point in the
same direction. Gravity acts downwards and displacement is also
downwards.

When two vectors point in the same direction, you can get the scalar
product by just multiplying the *value* of the two vectors together and
ignoring the direction.

But what happens if they don\'t point in the same direction?

Consider a man walking up a hill. Obviously it takes energy to do this
because you are going against the force of gravity. The steeper the
hill, the more energy it takes every step to climb it. This is something
we all know unless we live on a salt lake.

In a situation like this we can still work out the work done. In the
diagram, the green lines represent the displacement. To find out how
much work *against* gravity the man does, we work out the *projection*
of the displacement along the line of action of the force of gravity. In
this case it\'s just the y component of the man\'s displacement. This is
where the cos θ comes in. θ is merely the angle between the velocity
vector and the force vector.

If the two forces do not point in the same direction, you can still get
the scalar product by multiplying the projection of one force in the
direction of the other force. Thus:
`{{PSG/eq|<math>\vec{a}\cdot\vec{b}\equiv \|\vec{a}\|\ \|\vec{b}\|\ \cos\theta</math>}}`{=mediawiki}
There is another method of defining the dot product which relies on
components.

  --------------------------------------------------
   $\vec{a}\cdot\vec{b}\equiv a_xb_x+a_yb_y+a_zb_z$
  --------------------------------------------------

##### What is a \"cross-product\"? (Force on a charged particle in a magnetic field)

**Answer:** Suppose there is a charged particle moving in a constant
magnetic field. According to the laws of electromagnetism, the particle
is acted upon by a force called the Lorentz force. If this particle is
moving from left to right at 30 m/s and the field is 30 Tesla pointing
straight down perpendicular to the particle, the particle will actually
curve in a circle spiraling out of the plane of the two with an
acceleration of its charge in coulombs times 900 newtons per coulomb!
This is because the calculation of the Lorentz force involves a
cross-product.when cross product can replace the sin0 can take place
during multiplication. A cross product can be calculated simply using
the angle between the two vectors and your right hand. If the forces
point parallel or 180° from each other, it\'s simple: the cross-product
does not exist. If they are exactly perpendicular, the cross-product has
a magnitude of the product of the two magnitudes. For all others in
between however, the following formula is used:

  --------------------------------------------------------------------------------------------------
   $\left\|\vec{a}\times\vec{b}\right\| = \left\|\vec{a}\right\|\ \left\|\vec{b}\right\|\sin\theta$
  --------------------------------------------------------------------------------------------------

!The right-hand rule: point your index finger along the first vector
and your middle finger across the second; your thumb will point in the
direction of the resulting
vector
But if the result is a vector, then what is the direction? That too is
fairly simple, utilizing a method called the \"right-hand rule\".

The right-hand rule works as follows: Place your right-hand flat along
the first of the two vectors with the palm facing the second vector and
your thumb sticking out perpendicular to your hand. Then proceed to curl
your hand towards the second vector. The direction that your thumb
points is the direction that cross-product vector points! Though this
definition is easy to explain visually it is slightly more complicated
to calculate than the dot product.

  ---------------------------------------------------------------------------------------------
   $(a_x,\ a_y,\ a_z)\times(b_x,\ b_y,\ b_z) =(a_yb_z-a_zb_y\ ,a_xb_z-a_zb_x\ ,a_xb_y-a_yb_x)$
  ---------------------------------------------------------------------------------------------

##### How to draw vectors that are in or out of the plane of the page (or board)

!How to draw vectors in the plane of the
paper
!Standard symbols of a vector going into or out of a
page
**Answer:** Vectors in the plane of the page are drawn as arrows on the
page. A vector that goes into the plane of the screen is typically drawn
as circles with an inscribed X. A vector that comes out of the plane of
the screen is typically drawn as circles with dots at their centers. The
X is meant to represent the fletching on the back of an arrow or dart
while the dot is meant to represent the tip of the arrow.

