Solve the following equations using Cramer's rule (determinant) a) x−3y=−3, 2x+5y=8 b) x−2y-z=3, 3x−6y−5z=3…

Question

Solve the following equations using Cramer's rule (determinant) a) x−3y=−3, 2x+5y=8 b) x−2y-z=3, 3x−6y−5z=3, 2X-y+z=0 c) x+2y+3z=17, 3x+2y+z=11, x−5y+3z=−5

ScanSolve AI Tutor · Accepted Answer

Here's the solution for question 9:

To determine if vectors are orthogonal, we check if their dot product is zero. If they are orthogonal, we then check if they are orthonormal by verifying if both vectors are unit vectors (i.e., their magnitudes are 1).

i) Given vectors $a = (4, 2, -1)$ and $b = (1, -3, -2)$ .

Calculate the dot product $a \cdot b$ .
$a \cdot b = (4)(1) + (2)(-3) + (-1)(-2)$ $a \cdot b = 4 - 6 + 2$ $a \cdot b = 0$
Determine if the vectors are orthogonal.
Since $a \cdot b = 0$ , the vectors $a$ and $b$ are orthogonal.
Calculate the magnitudes of $a$ and $b$ .
$||a|| = \sqrt{4^2 + 2^2 + (-1)^2} = \sqrt{16 + 4 + 1} = \sqrt{21}$ $||b|| = \sqrt{1^2 + (-3)^2 + (-2)^2} = \sqrt{1 + 9 + 4} = \sqrt{14}$
Determine if the vectors are orthonormal.
Since $||a|| = \sqrt{21} \neq 1$ and $||b|| = \sqrt{14} \neq 1$ , the vectors are not unit vectors.
Therefore, the vectors $a$ and $b$ are not orthonormal.

ii) Given vectors $a = (5, -1, 3)$ and $b = (4, 7, -2)$ .
Calculate the dot product $a \cdot b$ .
$a \cdot b = (5)(4) + (-1)(7) + (3)(-2)$ $a \cdot b = 20 - 7 - 6$ $a \cdot b = 13 - 6$ $a \cdot b = 7$
Determine if the vectors are orthogonal.
Since $a \cdot b = 7 \neq 0$ , the vectors $a$ and $b$ are not orthogonal.
Determine if the vectors are orthonormal.
Since the vectors are not orthogonal, they cannot be orthonormal.

Final Answer:

i) The vectors $a = (4, 2, -1)$ and $b = (1, -3, -2)$ are orthogonal but not orthonormal.

ii) The vectors $a = (5, -1, 3)$ and $b = (4, 7, -2)$ are not orthogonal and therefore not orthonormal.

Solve the following equations using Cramer's rule (determinant) a) x-3y=-3, 2x+5y=8 b) x−2y−z=3, 3x−6y−5z=3, 2X−y+z=0 c) x+2y+3z=17, 3x+2y+z=11, x-5y+3z=-5

Step-by-step solution

Calculate the dot product $a \cdot b$ .

Determine if the vectors are orthogonal.

Calculate the magnitudes of $a$ and $b$ .

Determine if the vectors are orthonormal.

Calculate the dot product $a \cdot b$ .

Determine if the vectors are orthogonal.

Determine if the vectors are orthonormal.

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More Mathematics Questions

Solve the following equations using Cramer's rule (determinant) a) x-3y=-3, 2x+5y=8 b) x−2y−z=3, 3x−6y−5z=3, 2X−y+z=0 c) x+2y+3z=17, 3x+2y+z=11, x-5y+3z=-5

Step-by-step solution

Calculate the dot product a⋅ba \cdot ba⋅b.

Determine if the vectors are orthogonal.

Calculate the magnitudes of aaa and bbb.

Determine if the vectors are orthonormal.

Calculate the dot product a⋅ba \cdot ba⋅b.

Determine if the vectors are orthogonal.

Determine if the vectors are orthonormal.

Still stuck on this one?

More Mathematics Questions

Calculate the dot product $a \cdot b$ .

Calculate the magnitudes of $a$ and $b$ .

Calculate the dot product $a \cdot b$ .