Answer:
[tex]\textsf{(a)}\quad \overrightarrow{CB}=a-4b[/tex]
[tex]\textsf{(b)}\quad \displaystyle\overrightarrow{CB}=\binom{-1}{15}[/tex]
Step-by-step explanation:
Given vectors of triangle ABC:
[tex]\overrightarrow{AB}=a+5b[/tex]
[tex]\overrightarrow{AC}=9b[/tex]
Part (a)
To express vector CB in terms of a and b, we can use the triangle law of vector addition:
[tex]\overrightarrow{CB}=\overrightarrow{CA}+\overrightarrow{AB}[/tex]
Since [tex]\overrightarrow{CA}[/tex] is in the opposite direction to [tex]\overrightarrow{AC}[/tex], then [tex]\overrightarrow{CA}=-9b[/tex].
Therefore:
[tex]\overrightarrow{CB}=-9b+a+5b[/tex]
[tex]\overrightarrow{CB}=a-4b[/tex]
So, vector CB expressed in terms of a and b is:
[tex]\Large\boxed{\boxed{\overrightarrow{CB}=a-4b}}[/tex]
Part (b)
Given column vectors:
[tex]\displaystyle a=\binom{3}{7}\\\\\\b=\binom{1}{-2}[/tex]
Substitute these values into the expression for vector CB from part (a):
[tex]\displaystyle\overrightarrow{CB}=\binom{3}{7}-4\binom{1}{-2}[/tex]
[tex]\displaystyle\overrightarrow{CB}=\binom{3}{7}-\binom{4}{-8}[/tex]
[tex]\displaystyle\overrightarrow{CB}=\binom{3-4}{7-(-8)}[/tex]
[tex]\displaystyle\overrightarrow{CB}=\binom{-1}{15}[/tex]
So, the expression for vector CB as a column vector is:
[tex]\large\boxed{\boxed{\displaystyle\overrightarrow{CB}=\binom{-1}{15}}}[/tex]
