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Find the value of two 𝑧 plus 𝑖𝑧 bar given 𝑧 on the Argand diagram below.
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We know that any complex number 𝑧 can be written in the form 𝑎 plus 𝑏𝑖, where 𝑎 is the real part and 𝑏 the imaginary part of a complex number.
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On an Argand diagram, the real part or component is denoted by the horizontal axis and the imaginary part by the vertical axis.
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In this question, the point 𝑧 has coordinates four, five.
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Therefore, 𝑧 is the complex number four plus five 𝑖.
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The notation 𝑧 bar denotes the complex conjugate.
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We know that if 𝑧 is equal to 𝑎 plus 𝑏𝑖, then 𝑧 bar is equal to 𝑎 minus 𝑏𝑖.
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The imaginary part of our complex number has the opposite sign.
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This means that 𝑧 bar in our question is equal to four minus five 𝑖.
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This can be shown on the Argand diagram as the point with coordinates four, negative five.
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Note that the complex conjugate is a reflection in the real axis.
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We were asked to find the value of two 𝑧 plus 𝑖𝑧 bar.
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This is equal to two multiplied by four plus five 𝑖 plus 𝑖 multiplied by four minus five 𝑖.
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Distributing the first set of parentheses gives us eight plus 10𝑖.
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Distributing the second set gives us four 𝑖 minus five 𝑖 squared.
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Two 𝑧 plus 𝑖𝑧 bar is equal to eight plus 10𝑖 plus four 𝑖 minus five 𝑖 squared.
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From our knowledge of complex numbers, we know that 𝑖 squared is equal to negative one.
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Negative five multiplied by negative one is equal to five.
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Therefore, the right-hand side simplifies to eight plus 10𝑖 plus four 𝑖 plus five.
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We can then group or collect the real and imaginary parts separately.
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Eight plus five is equal to 13, and 10𝑖 plus four 𝑖 is 14𝑖.
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Two 𝑧 plus 𝑖𝑧 bar is equal to 13 plus 14𝑖.