Solve Lesson 8 Page 62 SBT Math 10 – Kite>


Topic

Find real numbers a and b such that each of the following pairs of vectors are equal:

a) \(\overrightarrow m = (2a + 3;b – 1)\) and \(\overrightarrow n = (1; – 2)\)

b) \(\overrightarrow u = (3a – 2;5)\)and \(\overrightarrow v = (5;2b + 1)\)

c) \(\overrightarrow x = (2a + b;2b)\) and \(\overrightarrow y = (3 + 2b;b – 3a)\)

Solution method – See details

\(\overrightarrow a = ({x_1};{y_1})\) and \(\overrightarrow b = ({x_2};{y_2})\) are equal if and only if \(\left\{ \begin{ array}{l}{x_1} = {x_2}\\{y_1} = {y_2}\end{array} \right.\)

Detailed explanation

a) \(\overrightarrow m = (2a + 3;b – 1)\) and \(\overrightarrow n = (1; – 2)\)

\(\overrightarrow m = \overrightarrow n \Leftrightarrow \left\{ \begin{array}{l}2a + 3 = 1\\b – 1 = – 2\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}a = – 1\\b = – 1\end{array} \right.\)

b) \(\overrightarrow u = (3a – 2;5)\)and \(\overrightarrow v = (5;2b + 1)\)

\(\overrightarrow u = \overrightarrow v \Leftrightarrow \left\{ \begin{array}{l}3a – 2 = 5\\5 = 2b + 1\end{array} \right. \Leftrightarrow \left\{ \ begin{array}{l}a = \frac{7}{3}\\b = 2\end{array} \right.\)

c) \(\overrightarrow x = (2a + b;2b)\) and \(\overrightarrow y = (3 + 2b;b – 3a)\)

\(\overrightarrow x = \overrightarrow y \Leftrightarrow \left\{ \begin{array}{l}2a + b = 3 + 2b\\2b = b – 3a\end{array} \right. \Leftrightarrow \left\ { \begin{array}{l}2a – b = 3\\3a + b = 0\end{array} \right \Leftrightarrow \left\{ \begin{array}{l}a = \frac{3} {5}\\b = – \frac{9}{5}\end{array} \right.\)



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