If a = 3i + 2j - k and b = -2i + j calculate each magnitude.
a) a + b
I got sqrt3, but the answer at the back of the book says sqrt11.
2) If D(3,4,5) and E(-2,1,5) are points in space calculate each expression and state what it represents.
|OD|
can you please do those and explain steps.
a + b = (3-2) i +(2+1) j +(-1+0) k
= 1 i + 3 j -1 k
sqrt (1^1 + 3^2 + (-1)^2 )
sqrt ( 1 + 9 + 1)
sqrt (11)
The book wins again :)
2) If D(3,4,5) and E(-2,1,5) are points in space calculate each expression and state what it represents.
|OD|
-------------------
I assume 0 is the origin (0,0,0)
so OD = (3-0)i + (4-0) j + (5-0) k
and
|OD| = sqrt (3^2 + 4^2 + 5^2)
=sqrt (9 + 16 + 25)
= sqrt(50)
= 5 sqrt(2)
Wow I accidentally put a - in front of 2 and ended up with -1. Thanks!
a) To calculate the magnitude of a vector, you need to find the length of the vector. The magnitude of a vector can be calculated using the Pythagorean theorem. In this case, we are given two vectors a and b. To find the magnitude of the vector a + b, we need to add the corresponding components of a and b and then calculate the magnitude of the resulting vector.
a = 3i + 2j - k
b = -2i + j
To find a + b, we add the corresponding coefficients of i, j, and k:
a + b = (3i + 2j - k) + (-2i + j) = i + 3j - k
To find the magnitude of a + b, we calculate the square root of the sum of the squares of the coefficients:
Magnitude of a + b = sqrt((1^2) + (3^2) + (-1^2)) = sqrt(1 + 9 + 1) = sqrt(11)
Therefore, the correct answer for the magnitude of a + b is sqrt(11), not sqrt(3).
b) To calculate the magnitude of a vector, we again use the Pythagorean theorem. In this case, we are given two points in space, D(3,4,5) and E(-2,1,5). To calculate the expression |OD|, we need to find the distance between point O(0,0,0) and point D(3,4,5).
To find the distance between two points in space, we can use the following formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2)
In this case, O(0,0,0) and D(3,4,5), so we substitute the values into the formula:
|OD| = sqrt((3 - 0)^2 + (4 - 0)^2 + (5 - 0)^2)
= sqrt(3^2 + 4^2 + 5^2)
= sqrt(9 + 16 + 25)
= sqrt(50)
Therefore, the expression |OD| represents the distance between the points O and D, which is sqrt(50).