ABCDEF is a regular hexagon. If AB=a,BC=b,express vector,AD AND AE in terms of a and b.
Join line AC to form tri.ABC and get AC=AB+BC
Join line AD to form tri.ACD and get AD=AB+BC+CD
To express vector AD and AE in terms of a and b, we need to understand the geometry of a regular hexagon and how vectors are defined in relation to its sides.
A regular hexagon has six equal sides, and each interior angle measures 120 degrees. Let's label the vertices of the hexagon as follows:
B_____C
/ \
A D
\ /
F_____E
In a regular hexagon, each interior angle subtends an arc equivalent to the length of the hexagon side. Therefore, angle ADB = 120 degrees, and angle AEB = 120 degrees as well.
To express vector AD in terms of a and b, we need to move in the counterclockwise direction from point A to point D. Since AB = a, we can represent vector AD as AD = AB + BD.
To find the vector BD, we can split it into two components: one along AB and the other perpendicular to AB. Let's call the component parallel to AB as vector BD₁ and the component perpendicular to AB as vector BD₂.
Since angle ADB = 120 degrees, angle ABD = 30 degrees (complementary to 90 degrees). Using basic trigonometry, we can relate BD₁ to AB:
BD₁ = AB * cos(30 degrees)
= a * cos(30 degrees)
The component BD₂ is perpendicular to AB, so it will have the same magnitude as BD₁, but in the opposite direction:
BD₂ = -BD₁
= -a * cos(30 degrees)
Therefore, vector BD = BD₁ + BD₂ = a * cos(30 degrees) - a * cos(30 degrees) = 0.
So, vector AD = AB + BD = a + 0 = a.
Therefore, vector AD = a.
Similarly, to find vector AE, we move in the counterclockwise direction from point A to point E. Since angle AEB = 120 degrees, angle ABE = 30 degrees (complementary to 90 degrees). Using basic trigonometry, we can relate AE to AB:
AE = AB * cos(30 degrees)
= a * cos(30 degrees)
Therefore, vector AE = a * cos(30 degrees).
In summary:
Vector AD = a
Vector AE = a * cos(30 degrees)
So, we have expressed vector AD and AE in terms of a and b.