Question:
Find (3a+b)dot(2b-4a) if a=-i-3j+k and b=2i+4j-5k
I did:
=3(-1,-3,1)+(2,4,-5)dot 2(2,4,-5)-4(-1,-3,1)
=(-3,-9,3)+(2,4,-5)dot(4,8,-10)-(4+12,-4)
=(-1,-5,-2)dot(0,-4,-6)
=(0,20,12)
=0+20+12
=32
The text book answer is -80. What did I do wrong/
oops. Flipped a sign:
=3(-1,-3,1)+(2,4,-5)dot 2(2,4,-5)-4(-1,-3,1)
=(-3,-9,3)+(2,4,-5)dot(4,8,-10)+(4+12,-4)
=(-1,-5,-2)dot(8,20,-14)
= -8-100+28
= -80
Sure looks like you corrected the typo, but made no effort to adjust the corresponding arithmetic.
http://www.jiskha.com/display.cgi?id=1423332996
The unnecessary duplication of work could have been avoided.
To find the dot product, you should multiply the corresponding components of the vectors and then add them up.
Given:
a = -i - 3j + k
b = 2i + 4j - 5k
Let's calculate the dot product step by step:
(3a + b) dot (2b - 4a)
= 3(-i - 3j + k) dot (2(2i + 4j - 5k) - 4(-i - 3j + k))
= (-3i - 9j + 3k) dot (4i + 8j - 10k + 4i + 12j - 4k)
= (-3i - 9j + 3k) dot (8i + 20j - 14k)
Now, let's calculate the dot product by multiplying and adding the corresponding components:
(-3i - 9j + 3k) dot (8i + 20j - 14k)
= (-3)(8) + (-9)(20) + (3)(-14)
= -24 - 180 - 42
= -246
Thus, the correct result is -246, not -80.
To find the dot product of two vectors, you need to multiply their corresponding components and then sum up the results. Let's go through the steps to find the correct answer:
Given:
a = -i - 3j + k
b = 2i + 4j - 5k
Step 1: Calculate (3a + b)
To find (3a + b), you need to multiply each component of a by 3 and each component of b by 1, and then add them together.
(3a + b) = 3(-i - 3j + k) + (2i + 4j - 5k)
= (-3i - 9j + 3k) + (2i + 4j - 5k)
= -3i + 2i - 9j + 4j + 3k - 5k
= -i - 5j - 2k
Step 2: Calculate (2b - 4a)
To find (2b - 4a), you need to multiply each component of b by 2 and each component of a by -4, and then subtract them.
(2b - 4a) = 2(2i + 4j - 5k) - 4(-i - 3j + k)
= (4i + 8j - 10k) - (-4i - 12j + 4k)
= 4i + 8j - 10k + 4i + 12j - 4k
= 8i + 20j - 14k
Step 3: Calculate the dot product using the formula
To calculate the dot product, you multiply the corresponding components and then sum them up.
(3a + b) · (2b - 4a) = (-i - 5j - 2k) · (8i + 20j - 14k)
= -i(8i) + (-i)(20j) + (-i)(-14k) +
(-5j)(8i) + (-5j)(20j) + (-5j)(-14k) +
(-2k)(8i) + (-2k)(20j) + (-2k)(-14k)
= -8i^2 - 20ij + 14ik - 40ij - 100j^2 + 70jk - 16ki - 40kj + 28k^2
= -8i^2 - 60ij + 70jk - 100j^2 + 14ik - 16ki + 28k^2
Now, you need to express the terms in terms of i, j, and k using the given values.
i^2 = -1, j^2 = -1, k^2 = -1
ij = ji = k, jk = kj = -i, ki = ik = -j
Substituting these values:
(3a + b) · (2b - 4a) = -8(-1) - 60k + 70(-i) - 100(-1) + 14k - 16(-i) + 28(-1)
= 8 + 60k - 70i + 100 + 14k + 16i - 28
= 108 - 54i + 74k
So, the correct answer is 108 - 54i + 74k. It seems your calculation mistake occurred during the step where you multiplied the terms incorrectly. Going through the steps correctly will help you get the correct answer of 108 - 54i + 74k.