Question

Given that r1=3i +2j+k, r2=2i-4j-3k, r3=-1+2j+2k. Find the magnitude of

a. R=r1+ r2 r3
B. A=2r1-3r2-5r3

Answer 1

A. To find the magnitude of R, we first need to find the vector R by adding the vectors r1, r2, and r3.

R = r1 + r2 + r3 = (3i + 2j + k) + (2i - 4j - 3k) + (-1i + 2j + 2k) = (4i - 2j + 2k)

The magnitude of R is given by the formula ∣R∣ = √(R⋅R), where R⋅R is the dot product of R with itself.

∣R∣ = √(R⋅R) = √(4i^2 + (-2j)^2 + (2k)^2) = √(16 + 4 + 4) = √(24) = 2√(6)

B. To find the magnitude of A, we first need to find the vector A by applying the scalar multiplication 2 to r1 and subtracting scalar multiplications of r2 and r3.

A = 2r1 - 3r2 - 5r3 = (6i + 4j + 2k) - (6i - 12j + 15k) - (-5i + 10j + 10k)

A = (6i + 4j + 2k) - (6i - 12j + 15k) - (-5i + 10j + 10k) = (11i + 2j - 13k)

The magnitude of A is given by the formula ∣A∣ = √(A⋅A), where A⋅A is the dot product of A with itself.

∣A∣ = √(A⋅A) = √(11i^2 + 2j^2 + (-13k)^2) = √(121 + 4 + 169) = √(294) = 2√(73)

Answer 2

To find the magnitude of R, add the components of r1, r2, and r3 and calculate the magnitude using the formula |R| = sqrt((Rx)^2 + (Ry)^2 + (Rz)^2). In this case, the magnitude of R is 4. To find the magnitude of A, add the components of 2r1, -3r2, and -5r3 and calculate the magnitude using the same formula. In this case, the magnitude of A is 14.

To find the magnitude of the vector sum R = r1 + r2 + r3:

R = (3i + 2j + k) + (2i - 4j - 3k) + (-1 + 2j + 2k)

R = (3 + 2 - 1)i + (2 - 4 + 2)j + (1 - 3 + 2)k

R = 4i + 0j + 0k

The magnitude of R is |R| = sqrt((4)^2 + (0)^2 + (0)^2)

|R| = sqrt(16) = 4

To find the magnitude of the vector sum A = 2r1 - 3r2 - 5r3:

A = 2(3i + 2j + k) - 3(2i - 4j - 3k) - 5(-1 + 2j + 2k)

A = (6i + 4j + 2k) - (6i - 12j - 9k) - (-5 + 10j + 10k)

A = 6i + 4j + 2k - 6i + 12j + 9k + 5 - 10j - 10k

A = 0i + 6j + k + 5 - 10j - 10k

A = -10i - 4j - 9k

The magnitude of A is |A| = sqrt((-10)^2 + (-4)^2 + (-9)^2)

|A| = sqrt(196) = 14