Hey you guys I have a midterm coming soon and I was just wondering how you guys

Question

Hey you guys I have a midterm coming soon and I was just wondering how you guys would go about solving these types of problems.

For example for the first part ,dot product is basically the x and y multipled with each other and then you add them.

This is what types of questions it contains:

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part question where you'll be given two vectors and and asked to compute: dot product, cross product, unit vector in the direction of one of them, scalar and vector projections of one onto the other Given 3 coplanar points, find a vector orthogonal to the plane containing them and something geometric about the points. Find equation of a plane parallel to and a given distance from a given plane Something about the relationship between vectors and the angle between them (not going to get anymore specific here) Distance formula for the distance between a given plane and a point not on the plane I will either give you the cross-sections (traces) of a quadric surface (or cylinder) and you have to give the equation of the surface or sketch the surface in R^3 or I'll give you the equation and you'll have to sketch the cross sections and then sketch the surface in the first octant.

Explanation / Answer

<a> . <b> = (axi+ayj+azk).(bxi+byj+bzk)

<a>x<b>=

| i j k |

| ax ay az |

| bx by bz |

unit vector

<c> =<a>/|a| unit vector in dir of <a>

<d>=<b>/|b| unit vector in dir of <b>

<e> = <a>x<b>/(|<a>x<b>|) unit vector in dir of <a>x<b>

to find equation of line

first find equation of plane

(X-ax)/b =(Y-ay)/c=(Z-az)/d

will give the equation

let any plane parallel to plane x-3y+z=-17 is

x-3y+z= d
it passes through A(2, 1, -- 3) so
2 -- 3(1) + (--3) = d Or d = -- 4
required plane is x --3y + z + 4 = 0

cos? = a.b / |a||b| to find angle betn 2 vectors

Given a plane

ax+by+cz+d=0

and a point x_0=(x_0,y_0,z_0), the normal vector to the plane is given by

v=[a; b; c],

and a vector from the plane to the point is given by

w=-[x-x_0; y-y_0; z-z_0].

Projecting w onto v gives the distance D from the point to the plane as

D = |proj_(v)w|

= (|v

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