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by xprofessor on September 16th, 2008
By 'hat' I assume you mean the symbol ^ , which when placed over i and j means UNIT VECTORS along two perpendicular axes. I'll write i^ and j^ to mean i-hat and j-hat.
So it's similar to plotting the point (-4,4) on the x-y Cartesian plane. The numbers -4 and +4 are the vector's COMPONENTS.
[edit]
The unit vectors i^ and j^ go from the origin(0,0) to the points (1,0) and (0,1), respectively. (Sometimes you use -1.)
The components are like a recipe. The vector from the origin (0,0) to the point (-4,4) is composed of -4 of i^ and +4 of j^. Since i^ = (1,0) then -4i^ = (-4,0) etc. When you take a vector of length 4 in the OPPOSITE direction of i^, and add it to a vector of length 4 ALONG the direction of j^, you get a vector at 45 degrees according to the parallelogram rule.
Put differently, the vector from (0,0) to (-4,4) DECOMPOSES into -4i^ + 4j^. (also see LINEAR COMBINATION)
The length of this vector is 4√2 using the Pythagorean theorem.
by krazymike on September 16th, 2008
hat? is that supposed to be -4i + 4j?
You simply put a point on (-4, 4) just like it's a Cartesian plane. The resultant magnitude is 4√2
by bobbybrown on September 16th, 2008
so, will the be along the X and Y axi? and how do you calculate the magnitude and direction, given VectorC = (-4i^, 4j^)
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You're reading With vector alegbra, how would you plot such a vector on a coordinate plane as (-4 i-hat + 4 j-hat)?
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Comments
so does the vector begin at the coordinate (-4, 4)? Also, how do i find the magnitude and direction of the vector only given that information?
by bobbybrown on September 16th, 2008
Both the unit vectors i and j begin at the origin and have length 1 unit. -4 i^ means multiply i^ by the real number scalar -4. So you get a vector of length 4 units pointing in the opposite direction of i^. The other vector is also of length 4 pointing along j^. All vectors here begin at the origin.
by xprofessor on September 16th, 2008
btw, in 3 dimensions they throw in a k^ !
by xprofessor on September 16th, 2008
so, would both vectors be along the X and Y axi? and how do you determine the magnitude and direction when given VectorC = (-4i^, 4j^)
by bobbybrown on September 16th, 2008
also in some contexts i^ and j^ are NOT necessarily perpendicular but must still be distinct (in different directions).
by xprofessor on September 16th, 2008
The final vector that goes from the origin to (-4,4) is midway between the axes or unit vectors. This vector is the sum of -4 i^-vectors and +4 j^-vectors). It's components are -4 of i^ and 4 of j^ -- like a recipe. Did you learn the parallelogram rule of adding vectors? Here's it's a square 4 units on a side. Actually I'm ASSUMING i^ and j^ are perpendicular. Are they?
by xprofessor on September 16th, 2008
all the directions i am given is:
draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction.
a. VectorB = -4i^ + 4j^
by bobbybrown on September 16th, 2008
I'm confused by 'label an angle'. You could always say 0 degrees and 90 degrees (or 0 and pi/2 radians). If the unit vectors are at, say, 30 deg and 60 deg, then you have to do some geometry or trig as well to compute the parallelogram, which will no longer be a square. Good luck!
by xprofessor on September 16th, 2008