A and B are vectors in the same plane. If A= 8i+6j. Find the vector B such that |A+B| = 14.14 units and A is perpendicular to B.
A is perpendicular to B
so
A dot B = 0
AxBx + AyBy = 0
8 Bx + 6 By = 0
and
A + B = (Ax + Bx)i+ (Ay+By)j
|A+B|^2 = 14.14^2 = (Ax+Bx)^2+(Ay+By)^2
so
(Ax+Bx)^2+(Ay+By)^2 = 100
(8+Bx)^2 +(6+By)^2 = 100
but we know
By =-(8/6)Bx = -(4/3) Bx
plug and chug.
Sorru for the inconvenience... but I didnt get this concept
|A+B|^2 = 14 ??!!
|A+B|^2 = 14.14^2 = 200 not 100
I did not use calculator, knew sqrt 2 = 1.414
Oh .. the rest of the steps are the same then ?
yes
To find vector B, we need to determine its magnitude and direction. Since |A + B| = 14.14 units, it means that the magnitude of A + B is equal to 14.14 units.
Let's assume vector B = xi + yj, where x and y are unknown scalars. Now, we can find the magnitude of A + B using the given information:
|A + B| = 14.14 units
|A + xi + yj| = 14.14 units
To find the magnitude of a vector (in this case A + B), we use the Pythagorean theorem:
|A + B| = sqrt((8 + x)^2 + (6 + y)^2) = 14.14 units
Now, since A is perpendicular to B, the dot product of A and B will be zero. The dot product of two vectors A = a1i + a2j and B = b1i + b2j is given by:
A · B = (a1 * b1) + (a2 * b2) = 0
Substituting the known values, we get:
(8 * x) + (6 * y) = 0
Now, we have two equations:
Equation 1: sqrt((8 + x)^2 + (6 + y)^2) = 14.14 units
Equation 2: (8 * x) + (6 * y) = 0
To solve these equations simultaneously, we can use substitution or elimination.
Substituting Equation 2 into Equation 1, we have:
sqrt((8 + x)^2 + (6 + y)^2) = 14.14 units
From Equation 2, we can express x in terms of y:
x = -(6/8) * y = -(3/4) * y
Now, we can substitute the expression for x into Equation 1:
sqrt((8 -(3/4) * y)^2 + (6 + y)^2) = 14.14 units
Expanding and simplifying, we have:
sqrt((64 - (48/4)y + (9/16)y^2) + (36 + 12y + y^2)) = 14.14 units
Simplifying further:
sqrt(64 - 12y + 9/16y^2 + 36 + 12y + y^2) = 14.14 units
Taking the square of both sides to eliminate the square root:
64 - 12y + 9/16y^2 + 36 + 12y + y^2 = (14.14)^2
Simplifying further:
16y^2 + y^2 - 12y + 12y + 64 + 36 - (14.14)^2 = 0
Combining like terms:
17y^2 + 100 - (14.14)^2 = 0
Solving this quadratic equation will give us the values of y. Once we have y, we can substitute it into Equation 2 to find x. These values of x and y will represent vector B.