UNIVERSITY OF SOUTH AFRICA
College of Science, Engineering and Technology
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
MAT1503: Linear Algebra
Assignment 04 | 2026
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
MAT1503
Module Code:
Linear Algebra
Module Name:
Vectors, Planes, Lines, and Complex Num-
Assignment Topic:
bers
04
Assignment Number:
31 August 2026
Due Date:
Submitted in partial fulfilment of the requirements for MAT1503, UNISA 2025
, UNISA | MAT1503 Linear Algebra: Assignment 04
Question 1: Vector Projection and the Intersection of Planes
The problem sets up two vectors ⃗u = ⟨1, 1, 0⟩ and ⃗v = ⟨0, 1, 1⟩, together with a third vector w
⃗
defined through the vector projection of ⃗a = ⟨2, 2, 1⟩ onto ⃗b = ⟨1, 0, −2⟩.
1.1 Determining w
⃗
The vector projection formula is
⃗a · ⃗b ⃗
proj⃗b ⃗a = b.
⃗b · ⃗b
The dot product in the numerator is
⃗a · ⃗b = (2)(1) + (2)(0) + (1)(−2) = 2 + 0 − 2 = 0.
Since ⃗a · ⃗b = 0, the projection collapses to the zero vector regardless of ⃗b · ⃗b:
proj⃗b ⃗a = ⃗0.
Geometrically, ⃗a and ⃗b are orthogonal, so ⃗a has no component lying along ⃗b. A projection
onto ⃗b therefore vanishes identically, and a vector with zero magnitude has no well-defined
direction. The condition that w⃗ "points in the direction of the projection” cannot be satisfied
√
simultaneously with the condition |w|⃗ = 12, since the only vector lying along a zero vector is
√
itself the zero vector, whose magnitude is 0, not 12.
proj⃗b ⃗a = ⃗0, so w
⃗ is not uniquely determined by the stated data.
The two conditions imposed on w
⃗ are therefore mutually inconsistent as the question is
phrased, since orthogonality between ⃗a and ⃗b removes any directional information for w
⃗ to
inherit.
Page 2 of 10
College of Science, Engineering and Technology
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
MAT1503: Linear Algebra
Assignment 04 | 2026
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄⋄
MAT1503
Module Code:
Linear Algebra
Module Name:
Vectors, Planes, Lines, and Complex Num-
Assignment Topic:
bers
04
Assignment Number:
31 August 2026
Due Date:
Submitted in partial fulfilment of the requirements for MAT1503, UNISA 2025
, UNISA | MAT1503 Linear Algebra: Assignment 04
Question 1: Vector Projection and the Intersection of Planes
The problem sets up two vectors ⃗u = ⟨1, 1, 0⟩ and ⃗v = ⟨0, 1, 1⟩, together with a third vector w
⃗
defined through the vector projection of ⃗a = ⟨2, 2, 1⟩ onto ⃗b = ⟨1, 0, −2⟩.
1.1 Determining w
⃗
The vector projection formula is
⃗a · ⃗b ⃗
proj⃗b ⃗a = b.
⃗b · ⃗b
The dot product in the numerator is
⃗a · ⃗b = (2)(1) + (2)(0) + (1)(−2) = 2 + 0 − 2 = 0.
Since ⃗a · ⃗b = 0, the projection collapses to the zero vector regardless of ⃗b · ⃗b:
proj⃗b ⃗a = ⃗0.
Geometrically, ⃗a and ⃗b are orthogonal, so ⃗a has no component lying along ⃗b. A projection
onto ⃗b therefore vanishes identically, and a vector with zero magnitude has no well-defined
direction. The condition that w⃗ "points in the direction of the projection” cannot be satisfied
√
simultaneously with the condition |w|⃗ = 12, since the only vector lying along a zero vector is
√
itself the zero vector, whose magnitude is 0, not 12.
proj⃗b ⃗a = ⃗0, so w
⃗ is not uniquely determined by the stated data.
The two conditions imposed on w
⃗ are therefore mutually inconsistent as the question is
phrased, since orthogonality between ⃗a and ⃗b removes any directional information for w
⃗ to
inherit.
Page 2 of 10
