Bisecting Vectors: Find Unit Vector

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In summary, a bisecting vector is a vector that divides another vector into two equal parts. To find the unit vector of a bisecting vector, you need to find the midpoint of the original vector and divide it by two. This unit vector is useful in various applications and can be in any direction. The formula for finding the unit vector is u = (1/2)|v|v/|v|.
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Homework Statement



find a unit vector that bisects the angle between the vectors j+2k and 3i-j+k. report to 2 sig figs

Homework Equations





The Attempt at a Solution

 
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But how do you think you have to start?
 
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To find the unit vector that bisects the angle between the given vectors, we will first find the angle between them using the dot product formula. The dot product of two vectors is defined as the product of their magnitudes and the cosine of the angle between them.

Let vector A = j+2k and vector B = 3i-j+k.

Then, A · B = |A| |B| cosθ

= √(1^2+2^2) √(3^2+(-1)^2+1^2) cosθ

= √5 √11 cosθ

= √55 cosθ

To find the angle θ, we will use the inverse cosine function.

θ = cos^-1 (A · B / √(55))

= cos^-1 (√55 cosθ / √55)

= cos^-1 (cosθ)

= θ

Therefore, the angle between the given vectors is θ = cos^-1 (√55 cosθ / √55) = 45°.

To find the unit vector that bisects this angle, we will use the half-angle formula:

cosθ/2 = ±√((1+cosθ)/2)

cos(45°/2) = ±√((1+cos45°)/2)

= ±√((1+√2/2)/2)

= ±√((2+√2)/4)

= ±√(1+√2)/2

= ±0.9238795325

Thus, the unit vector that bisects the angle between the given vectors is ±0.9238795325 (i+2j+k). This can be rounded to 2 significant figures as ±0.92 (i+2j+k).
 

Related to Bisecting Vectors: Find Unit Vector

1. What is a bisecting vector?

A bisecting vector is a vector that divides another vector into two equal parts. It passes through the midpoint of the original vector and creates two equal angles.

2. How do you find the unit vector of a bisecting vector?

To find the unit vector of a bisecting vector, you first need to find the midpoint of the original vector. Then, calculate the length of the original vector and divide it by two. Finally, divide the original vector by the length you just calculated to get the unit vector of the bisecting vector.

3. What is the purpose of finding the unit vector of a bisecting vector?

The unit vector of a bisecting vector is useful in many applications, such as physics and engineering. It can be used to calculate the direction of a force or the direction of an object's movement.

4. Can a bisecting vector be in any direction?

Yes, a bisecting vector can be in any direction as long as it divides the original vector into two equal parts. It is not limited to any specific direction.

5. Is there a formula for finding the unit vector of a bisecting vector?

Yes, the formula for finding the unit vector of a bisecting vector is:
u = (1/2)|v|v/|v|
where u is the unit vector, v is the original vector, and |v| represents the length of the original vector.

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