Question

1. compute the dot product.
2. $langle 2, 3

angle cdot langle -5, 6
angle$

1. $langle 2, 3, 4

angle cdot langle -5, 6, -7
angle$

1. $langle -1, -8

angle cdot langle -4, 3
angle$

1. $(2hat{i} - 3hat{j}) cdot (4hat{j} - 7hat{k})$
2. $(3hat{i} - 2hat{j} + 4hat{k}) cdot (hat{i} + 5hat{j} - 6hat{k})$

Explanation:

Step1: Dot product formula for 2D vectors

For $\langle a,b
angle \cdot \langle c,d
angle = ac + bd$

Step1.1: Calculate first 2D dot product

$\langle 2,3
angle \cdot \langle -5,6
angle = (2)(-5) + (3)(6) = -10 + 18 = 8$

Step1.2: Calculate third 2D dot product

$\langle -1,-8
angle \cdot \langle -4,3
angle = (-1)(-4) + (-8)(3) = 4 - 24 = -20$

Step2: Dot product formula for 3D vectors

For $\langle a,b,c
angle \cdot \langle d,e,f
angle = ad + be + cf$

Step2.1: Calculate second 3D dot product

$\langle 2,3,4
angle \cdot \langle -5,6,-7
angle = (2)(-5) + (3)(6) + (4)(-7) = -10 + 18 - 28 = -20$

Step3: Dot product for unit vectors

$\hat{i} \cdot \hat{i}=1, \hat{j} \cdot \hat{j}=1, \hat{k} \cdot \hat{k}=1$; cross terms = 0

Step3.1: Calculate fourth unit vector dot product

$(2\hat{i} - 3\hat{j}) \cdot (4\hat{j} - 7\hat{k}) = (2)(0) + (-3)(4) + (0)(-7) = 0 -12 + 0 = -12$

Step3.2: Calculate fifth unit vector dot product

$(3\hat{i} - 2\hat{j} + 4\hat{k}) \cdot (\hat{i} + 5\hat{j} - 6\hat{k}) = (3)(1) + (-2)(5) + (4)(-6) = 3 -10 -24 = -31$

Answer:

1. $8$
2. $-20$
3. $-20$
4. $-12$
5. $-31$