Mathematics: (b) find real numbers a and b such that (a, 2, b.is orthogonal to both (−1, −2, 3) and (3, −2, 6).

You need to find GCD first it means greatest common divisor so GCD of (28,35) is 7 because 4x7=28 and 5x7=35 4+5=9 answer is nine...

(b) find real numbers a and b such that (a, 2,

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iixyloa

12.03.2022

Mathematics

Two vectors are orthogonal if their scalar product is zero. The scalar product of two vectors is computed as follows:

$(a_1,a_2,\ldots,a_n) \cdot (b_1,b_2,\ldots,b_n) = a_1b_1 + a_2b_2 + \ldots + a_nb_n$

In other words, you multiply each component of the first vector with the correspondant component of the second vector, and then you sum all these products.

So, we need to compute the two scalar products, and set them to zero:

$(a, 2,b) \cdot (-1, -2, 3) = 0 \iff -a-4+3b = 0$

$(a, 2,b) \cdot (3, -2, 6) = 0 \iff 3a-4+6b = 0$

We can rewrite the two requests in this linear system:

$\begin{cases} -a+3b = 4\\3a+6b = 4 \end{cases}$

From the first equation we can deduce a = 3b-4 , which substituted in the second gives

$3a+6b = 4 \to 3(3b-4)+6b = 4 \to 9b-12+6b = 4 \to b = \dfrac{16}{15}$

If we substitute this value for b in the first equation we have

$-a+3b = 4 \to -a+3\cdot \dfrac{16}{15} = 4 \to -a+\dfrac{16}{5} = 4 \to a = \dfrac{16}{5}-4= \dfrac{-4}{5}$

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(b) find real numbers a and b such that (a, 2, b.is orthogonal to both (−1, −2, 3) and (3, −2, 6).

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