Calculate the area of the parallelogram with sides consisting of the vectors:
a = (1,2,-2)
b = (-1,3,0)
Area of parallelogram formed by two vectors is the magnitude of the cross product.
Cross product=
i j k
1 2 -2
-1 3 0
=(2*0-(-2)*3)i
-(1*0-(-2)(-1))j
+(1*3-2(-1))k
=6i-2j+5k
Magnitude
=√(6²+(-2)²+5²)]
=√(65)
thank you!
You're welcome!
To calculate the area of a parallelogram using vectors, we can use the cross product of the vectors that represent its sides.
In this case, we have vectors a = (1, 2, -2) and b = (-1, 3, 0) as the sides of the parallelogram.
To find the cross product, we can use the following formula:
a x b = (aybz - azby, azbx - axbz, axby - aybx)
Let's calculate the cross product:
ax = 1, ay = 2, az = -2
bx = -1, by = 3, bz = 0
axb = (2*0 - -2*3, -2*-1 - 1*0, 1*3 - 2*-1)
= (0 - (-6), 2 - 0, 3 + 2)
= (6, 2, 5)
Now we have the cross product a x b = (6, 2, 5).
The magnitude of the cross product represents the area of the parallelogram formed by the two vectors.
To calculate the magnitude, we use the formula:
|a x b| = sqrt((axb)^2 + (ayb)^2 + (azb)^2)
Plugging in the values:
|a x b| = sqrt(6^2 + 2^2 + 5^2)
= sqrt(36 + 4 + 25)
= sqrt(65)
Therefore, the area of the parallelogram formed by vectors a and b is sqrt(65) square units.